Jie Liu
email: jie.liu@centralesupelec.fr

Enrico Zio
email: enrico.zio@centralesupelec.fr

A SVR-based ensemble approach for drifting data streams with recurring patterns

Keywords: Online learning, ensemble, feature selection, pattern drifts, Support Vector Regression (SVR), Feature Vector Selection (FVS), Reactor Coolant Pump (RCP), seal leakage, Nuclear Power Plant

Pattern drift is a common issue for machine learning in real applications, as the distribution generating the data may change under nonstationary environmental/operational conditions. In our previous work, a strategy based on Feature Vector Selection (FVS) has been proposed for enabling a Support Vector Regression (SVR) model to adaptively update with streaming data, but the proposed strategy suffers from the incapability of treating recurring patterns. An instance-based online learning approach is proposed in this paper, which can adaptively update an SVR-based ensemble model with steaming data points. The proposed approach reduces the computational complexity of the updating process by selecting only part of the newly available data and allows following timely the ongoing patterns by resorting to FVS. The proposed approach creates new submodels directly from a basic model and the sub-models represent separately the data stream at different periods. A dynamic ensemble selection strategy is integrated in the approach to select the sub-models most relevant to the new data point for deriving the prediction, while reducing the influence of the irrelevant ones. The weights of the different models in the ensemble are updated, based on their prediction errors. Comparison results with several benchmark approaches on several synthetic datasets and on the dataset concerning the leakage from the first seal in a Reactor Coolant Pump, prove the efficiency and accuracy of the proposed online learning ensemble approach.

Introduction

Predicting malfunctions and failures is a fundamental task for the reliable and safe operation of components and systems. Building an efficient and accurate predictor from available data is one of the main objectives in machine learning. This is particularly critical in high-risk industries, like nuclear, chemical process, aerospace and aviation ones. Regarding the nuclear industry, from which the case study of this paper is taken, most of the current approaches for prediction of parameters in NPP components and systems, are developed for static environments only [START_REF] Na | Prediction of Major Transient Scenarios for Severe Accidents of Nuclear Power Plants[END_REF], [START_REF] Kim | Application of neural networks to signal prediction in nuclear power plant[END_REF], [START_REF] Kim | error prediction for a nuclear-power-plant fault-diagnostic adviser using neural networks[END_REF], i.e. the data are assumed to originate from a process whose underlying distribution does not change with time. On the contrary, in practical applications, the underlying, normally unknown distribution generating the data can vary with time, causing pattern drifts. Pattern drifts in NPP components and systems can be due to a natural evolution of the environment, changes in the operational conditions or faults [START_REF] Alippi | Intelligence for Embedded Systems[END_REF]. In such cases, the data originate from nonstationary environments and models are developed and trained for static environments can no longer give accurate predictions for the new data. Pattern drifts can be divided into sudden pattern drifts, gradual pattern drifts and recurring patterns. Different approaches have been developed for tackling the different pattern drifts problems, which can be categorized into adaptive single model [START_REF] Lee | An incremental adaptive neural network model for online noisy data regression and its application to compartment fire studies[END_REF], [START_REF] Crammer | Online Passive-Aggressive Algorithms[END_REF], [START_REF] Csató | Sparse on-line gaussian processes[END_REF] and online learning ensembles [START_REF] Oza | Online ensemble learning[END_REF], [START_REF] Muhlbaier | An ensemble approach for incremental learning in nonstationary environments[END_REF], [START_REF] Minku | DDD: A new ensemble approach for dealing with concept drift[END_REF]. The former approach is based on an adaptive model that learns incrementally the new patterns and/or forgets the old inefficient ones; however in practice, the computational burden for incremental learning is unacceptable for large datasets, and the recurring patterns are not efficiently handled if they have already been deleted from the model. The online learning ensemble approach aims at updating an ensemble by adapting the sub-models weights and/or adding/deleting a submodel in the ensemble. The work reported in this paper focuses on this latter approach of online learning ensembles. There are different types of approaches for online learning ensembles, e.g. data-chunk-based approaches [START_REF] Erdem | Ensemble of SVMs for Incremental Learning[END_REF], [START_REF] Brzezinski | Reacting to different types of concept drift: the Accuracy Updated Ensemble algorithm[END_REF], [START_REF] Razavi-Far | Dynamic Weighting Ensembles for Incremental Learning and Diagnosing New Concept Class Faults in Nuclear Power Systems[END_REF], drift detector-based approaches, [START_REF] Kuncheva | Classifier Ensembles for Changing Environments[END_REF], [START_REF] Minku | The impact of diversity on online ensemble learning in the presence of concept drift[END_REF], [START_REF] Page | Continuous inspection schemes[END_REF], [START_REF] Baena-Garcia | Early Drift Detection Method[END_REF] instance-based approaches, [START_REF] Zliobaite | Active learning with drifting streaming data[END_REF], [START_REF] Kolter | Using additive expert ensembles to cope with concept drift[END_REF], [START_REF] Kadlec | Adaptive local learning soft sensor for inferential control support[END_REF] etc. Accuracy Weighted Ensemble (AWE) [START_REF] Ouyang | Mining Concept-Drifting and Noisy Data Streams Using Ensemble Classifiers[END_REF], Streaming Ensemble Algorithm (SEA) [START_REF] Street | A streaming ensemble algorithm (SEA) for large-scale classification[END_REF], Learning++.NSE [START_REF] Muhlbaier | An ensemble approach for incremental learning in nonstationary environments[END_REF] are some of the data-chunk-based approaches proposed in the literature. Learning++.NSE trains a new sub-model on the new data chunk if the prediction error exceeds a predefined threshold, and combines the sub-models built through a dynamically modified weighted majority voting. The sub-models weights are calculated based on their weighted-sum performance on different data chunks and added to the ensemble. The problem with these data chunk-based approaches is the determination of the size of the data chunks, as bigger chunks give more stable sub-models but different drifts may be contained in a single sub-model, whereas smaller chunks can better separate different drifts but lead to worse sub-models. There is also a delay in the ensemble for following the ongoing patterns, as the ensemble is updated only when a new data chunk is available and the patterns in the ensemble at this time may no longer be the ongoing patterns. In order to overcome these difficulties, various approaches have been proposed in the literature, which may combine a drift detector with online learning ensemble to alarm the need for a new submodel. Adaptive Classifier Ensemble (ACE) [START_REF] Nishida | ACE: Adaptive Classifiers-Ensemble system for concept-drifting environments[END_REF] slowly builds a new sub-model when the submodels error on the new data reaches a certain threshold. In [START_REF] Kuncheva | Classifier Ensembles for Changing Environments[END_REF], pattern drifts are detected by measuring the normalized weighted average output of the sub-models in the ensemble. Diversity analysis is used in [START_REF] Minku | The impact of diversity on online ensemble learning in the presence of concept drift[END_REF] to divide different drifts. The most popular drift detector algorithm is the Drift Detection Method (DDM) [START_REF] Page | Continuous inspection schemes[END_REF], which models the prediction error on each data point according to a binominal distribution. A new approach for online learning ensembles, called Diversity for Dealing with Drift (DDD) is proposed in [START_REF] Minku | DDD: A new ensemble approach for dealing with concept drift[END_REF], which maintains ensembles with different diversity levels. The experimental results show that DDD gives robust and accurate results. Although the drift detector-based approaches can solve the difficulty in deciding a good size of the data chunk, they, compared to instance-based updating approaches, still cannot update the ensemble once a pattern drift occurs, i.e. sufficient new data are needed before detecting and reacting to the pattern drifts. In [START_REF] Zliobaite | Active learning with drifting streaming data[END_REF], a theoretically supported framework for active learning of drifts in data streams is presented and three active learning strategies are developed based on separate uncertainty, dynamic allocation of labeling efforts over time and randomization of search space. Another instance-based approach, named Online Weighted Ensemble (OWE) is proposed in [START_REF] Soares | An on-line weighted ensemble of regressor models to handle concept drifts[END_REF] to learn new data points incrementally in the presence of different types of pattern drifts and to retain old information in recurring patterns. The instance-based updating approaches can learn the pattern drifts effectively and efficiently once they occur. But one main disadvantage is the computational complexity of updating the ensemble with every new data point. Furthermore, a dynamically weighted ensemble is proposed in [START_REF] Li | Mining Recurring Concept Drifts with Limited Labeled Streaming Data[END_REF] to store only the features most relevant to the learnt concept, which in turn increases the memory efficiency.

It is important to stress that online learning requires timely updating the ensemble, including its weights and sub-models, while minimizing to the extent feasible the computational burden brought by the updating operations needed for dealing with the different types of drifts. In this paper, an instance-based online learning approach for Support Vector Regression (SVR)based ensembles, named Online Ensemble based on Feature Vectors (noted OE-FV, for short), is proposed based on the Feature Vector Selection (FVS) approach presented in [START_REF] Baudat | Feature vector selection and projection using kernels[END_REF]. For convenience, in this paper, the term "input vector" refers to the vector containing all the input variables and "input" represents one input variable in the input vector. The term "pattern" refers to one input vectoroutput pair. SVR-based ensembles are made of sub-models trained with SVR. FVS calculates the geometrical linear relation among different input vectors of the data points in the Reproduced Kernel Hilbert Space (RKHS) and selects a small part of them as Feature Vectors (FVs) representing the whole training dataset. In our previous work [START_REF] Liu | An Adaptive Online Learning Approach for Support Vector Regression[END_REF], an adaptive online learning approach, named Online-SVR-FID has been proposed for a single SVR model to effectively follow the ongoing patterns by adjusting to two types of drifts (new pattern if the new input vector cannot be represented by a linear combination of the FVs in the model and changed pattern if the new input vector can be represented by a linear combination of the selected FVs but the prediction error is larger than a predefined threshold) and taking the correspondent action. If a new data point is judged as new pattern, it is added directly into the model, while if it is judged as a changed pattern, it is used to replace a selected pattern that makes least contribution to the recent updated models. Compared to several benchmark approaches, Online-SVR-FID has been shown to give comparable results while using much less time. One drawback of Online-SVR-FID is that the old patterns are deleted from the model and one needs to relearn the recurring patterns from scratch. Based on this previous work, an online learning ensemble is grown from the first kernel-based submodel 𝑀1 to store all the past patterns detected in the data, and each sub-model covers patterns in a certain period of the data stream. The ensemble is created sequentially by applying an online learning approach similar to Online-SVR-FID on 𝑀1. The online learning approach assures that the first sub-model 𝑀1 follows always the ongoing patterns. If the sub-models 𝑀1 for different periods of the data stream are separately saved and used in an ensemble, each of them is like an adjusted "copy" of 𝑀1 tailored to different instances of the online learning process. Every new sub-model is saved by copying the current 𝑀1 at the time when an old pattern risks of being deleted from the ensemble, as the process for updating 𝑀1 may use new data points to replace an existing pattern in 𝑀1 when the new data point is judged as a changed pattern. If the pattern to be replaced is unique in the ensemble and there is no more such pattern once deleted, the model before the replacement is copied and stored as a new sub-model to guarantee that all the occurred patterns appear at least once in the sub-models of the ensemble, and, then, the updated 𝑀1 is still the upto-date sub-model that continues to be updated with future new data points. Note that the sub-models are created sequentially and automatically from 𝑀1 and are not updated with the new data points. Through the FVS, only data points that are judged as new and changed patterns are used to update 𝑀1 and create new sub-models when the criterion is reached. Note that each time 𝑀1 is updated with a new or changed pattern, it is retrained to minimize the MSE on the recent data points. Thus, the computational burden bothering the instance-based approaches for online learning ensemble is reduced. The sub-models weights are updated with each new data point according to the weighted sum of the prediction errors on all the data points, where the prediction errors on the new data points are more weighted than the old ones. Thus, the ensemble can follow efficiently the ongoing patterns. Inspired by the work in [START_REF] Ko | From dynamic classifier selection to dynamic ensemble selection[END_REF], [START_REF] Cavalin | Dynamic selection approaches for multiple classifier systems[END_REF] and [START_REF] Alrifai | Combining global optimization with local selection for efficient QoS-aware service composition[END_REF], a dynamic ensemble selection strategy is also integrated in the proposed OE-FV. For each new data point, only the most relevant sub-models are used to form an ensemble and derive the weighted-sum prediction result, in order to avoid the influence of the poor ones. The dynamic selection of the sub-models are based on the geometric relation between the input vector of the new data point and the data points in each sub-model. Only the sub-models which can well represent the new input vector are selected. In order to test the efficiency and accuracy of OE-FV, experiments on several synthetic datasets and a real case study concerning the condition of a component of a Nuclear Power Plant (NPP) are carried out. Comparisons with Learn++.NSE and OWE show that the proposed approach gives better results in the experiments than those of OWE and Learn++.NSE, and the computation time of OE-FV is shorter than that of OWE. The rest of the paper is structured as follows. FVS and Online-SVR-FID are briefly reviewed in Section 2. Section 3 explains the approach proposed to build the ensemble automatically and the process of weights updating. The experiments on several synthetic datasets and a real case study regarding a component of a NPP are illustrated in Section 4. Comparisons with Learn++.NSE and OWE are also reported in this section. Some conclusions are drawn in Section 5.

Brief introduction of FVS and Online-SVR-FID

The proposed online learning approach for kernel-based ensemble is based on the work in [START_REF] Baudat | Feature vector selection and projection using kernels[END_REF] and [START_REF] Liu | An Adaptive Online Learning Approach for Support Vector Regression[END_REF]. In order to thoroughly explain the process of building and updating an ensemble with OE-FV, FVS [START_REF] Baudat | Feature vector selection and projection using kernels[END_REF] and Online-SVR-FID [START_REF] Liu | An Adaptive Online Learning Approach for Support Vector Regression[END_REF] are firstly and briefly reviewed in this section.

Feature Vector Selection

Suppose 𝑻 = {(𝒙 𝑖 , 𝑦 𝑖 ): 𝑖 = 1, 2, … , 𝑀} is the dataset at hand. FVS analyzes the geometric relation among the input vectors of different data points in a high-dimensional space, i.e. RKHS, and selects as FVs the ones which represent the dimensions of the RKHS related to the dataset, in order to decrease its complexity. The other input vectors in the dataset can be represented by a linear combination of the selected FVs in the RKHS. A model can be trained on the selected FVs with classical machine learning methods, e.g. SVR. In this paper, 𝝋(𝒙) is the mapping that maps an input vector from the original space to the RKHS and 𝑘(𝒙 𝑖 , 𝒙 𝑗 ) is the kernel function that represents the inner product 〈𝝋(𝒙 𝑖 ), 𝝋(𝒙 𝑗 )〉 in the RKHS. Once a new data point (𝒙 𝒏 , 𝑦 𝑛 ) with mapping 𝝋 𝒏 is available, we need to judge if this new data point is a new FV. Suppose the existing FVs selected form the dataset are {𝒙 1 , 𝒙 2 , … , 𝒙 𝐿 } and their mapping are included in the feature space 𝐒 = {𝝋 1 , 𝝋 2 , … , 𝝋 𝐿 }; the verification of a new FV amounts to finding the vector 𝒂 = {𝑎 1 , 𝑎 2 , … , 𝑎 𝐿 }, which gives the minimum of Equation ( 1) below:

𝜇 𝑛 = ‖𝝋 𝒏 -∑ 𝑎 𝑖 𝝋 𝑖 𝐿 𝑖=1 ‖ ‖𝝋 𝒏 ‖ . ( 1
)
Normally in kernel methods, it is difficult to know the exact expression of the mapping function 𝝋(𝒙). But kernel functions which represent the inner product between two mappings in RKHS can give a solution to the minimum of Equation ( 1). The minimum of 𝜇 𝑛 can be written as:

min 𝜇 𝑛 = 1 - 𝐾 𝑺,𝒏 𝑡 𝐾 𝑺,𝑺 -1 𝐾 𝑺,𝒏 𝑘(𝒙 𝒏 ,𝒙 𝒏 ) , (2)
where 𝐾 𝑺,𝑺 is the kernel matrix (gram matrix) of 𝐒 and 𝐾 𝑺,𝒏 = (𝑘(𝒙 𝑖 , 𝒙 𝒏 )), 𝑖 = 1,2, … , 𝐿 is the vector of the inner product between 𝝋 𝒏 and S. The derivation of Equation ( 2) from Equation ( 1) can be found in [START_REF] Baudat | Feature vector selection and projection using kernels[END_REF].

The value calculated with Equation ( 3) is called local fitness; the definition of global fitness is given in Equation ( 4); the vector 𝒂 can be calculated by Equation ( 5):

𝐽 𝑆 (𝒙 𝒏 ) = 𝐾 𝑺,𝒏 𝑡 𝐾 𝑺,𝑺 -1 𝐾 𝑺,𝒏 𝑘(𝒙 𝒏 ,𝒙 𝒏 ) (3)
𝐽 𝑺 = ∑ 𝐽 𝑺 (𝒙 𝑖 ) 𝑀 𝑖=1 (4) 𝒂 = 𝐾 𝑺,𝒏 𝑡 𝐾 𝑺,𝑺 -1 (5)
According to the definition of local fitness, each data point is called a pattern and the pattern drifts in this paper are divided into two types: new pattern if the new data point cannot be well represented by the existing FVs in any sub-model, i.e. 1 -𝐽 𝑆 (𝒙 𝒏 ) > 𝜌, with 𝜌 a small positive value; changed pattern if the new data point can be represented by the existing FVs in some sub-models, but the predicted value given by all the sub-models is not accurate enough, i.e. 1 -𝐽 𝑆 (𝒙 𝒏 ) < 𝜌 & |𝑦 ̂𝑛 -𝑦 𝑛 | > 𝜃 for all sub-models, with 𝑦 ̂𝑛 being the predicted value given by one current submodel and 𝜃 a positive value representing the tolerance on the prediction error.

Online-SVR-FID

Online-SVR-FID proposed in [START_REF] Liu | An Adaptive Online Learning Approach for Support Vector Regression[END_REF] aims at providing an efficient online learning approach for a single SVR model, based on FVS. This approach can be divided into two parts: offline training and online learning. The offline training is aimed at selecting FVs from the training dataset and training a model on the selected FVs, with the objective of minimizing the Mean Squared Error (MSE) on the whole training dataset.

In [START_REF] Liu | An Adaptive Online Learning Approach for Support Vector Regression[END_REF], it is showed that the number of selected FVs is very critical, as too few FVs cause low accuracy on the test dataset, while too many FVs cause overfitting the training dataset. The online learning is aimed at detecting the pattern drifts and taking corresponding reactions to update the model. If a new data point is a new FV, it is added to the model and the model is updated; if it is not a new FV and its prediction error is larger than the threshold 𝜃, it replaces the FV which makes least contribution to the recent models.

The case studies in [START_REF] Liu | An Adaptive Online Learning Approach for Support Vector Regression[END_REF] show that Online-SVR-FID gives comparable or even better results than the SVR trained on the whole training dataset, and Online-SVR-FID uses much less time for online learning the same test dataset. Comparisons with other popular online learning methods are also carried out with respect to the case studies and Online-SVR-FID gives always comparable accuracy.

Although the feature selection process increases the computational complexity for data preprocessing, the time for tuning hyperparameters in Online-SVR-FID is much less than the classic SVR model using the whole training dataset, as the Online-SVR-FID model is much less complicated.

Constrained by the length of the paper, the pseudo-code of Online-SVR-FID and the calculation of the contribution of each FV to the recent models are shown in Appendix. Note that Online-SVR-FID can make the model efficiently follow the ongoing patterns in the data. A main drawback of Online-SVR-FID is that some useful FVs are replaced during the UPDATE process shown in Appendix. Once these replaced patterns recur in the new data, the model needs to relearn them, i.e. the information in the past data are not fully stored in the single model.

To overcome this problem, in this paper an ensemble approach is proposed to store all the past patterns in the data and make a reasonable choice of sub-models when facing recurring patterns.

Fig. 1 The main procedure of OE-FV.

Fig. 2 Graphical updating process of OE-FV.

Training of the first sub-model in the ensemble (step 1 of Figure 1)

The first sub-model 𝑀 1 is trained on the training dataset, which is also the first and basic submodel in the ensemble (step 1 of Figure 1). In order to reduce the model complexity and computational burden, all the training dataset are not directly used to train the first sub-model.

Calculation of the predicted value of a new data point (step 2.1 of Figure 1)

For any new data point coming, prediction is obtained by a dynamic ensemble selection strategy (step 2.1 of Figure 1). A dynamic ensemble selection, as presented in [START_REF] Ko | From dynamic classifier selection to dynamic ensemble selection[END_REF], [START_REF] Cavalin | Dynamic selection approaches for multiple classifier systems[END_REF] and [START_REF] Alrifai | Combining global optimization with local selection for efficient QoS-aware service composition[END_REF], amounts to selecting the sub-models that are most relevant to the new data point and calculating their separate predictions; then, these predictions are fused by a weighted sum to give the final prediction of the ensemble for the new data point.

The dynamic selection of sub-models can be based on the overall local accuracy, local sub-model accuracy, a priori selection or a posteriori selection [START_REF] Ko | From dynamic classifier selection to dynamic ensemble selection[END_REF]. In OE-FV, the sub-models are selected on the basis of the local fitness of the new data point to the FVs of each sub-model, calculated by Equation ( 3). Only the sub-models with a local fitness that satisfies 1 -𝐽 𝑆𝑖 (𝒙) < 𝜌 are selected to form the ensemble predictor 𝐸𝑜𝐶 for the new data point. Suppose 𝐄𝐫 is the vector that contains the cumulated prediction errors of all the sub-models and 𝐄𝐫 𝐸𝑜𝑐 , which is a subset of 𝐄𝐫, contains the prediction errors of the sub-models in 𝐸𝑜𝐶: the weights of the selected sub-models are calculated as Equation ( 6) below:

𝛚 = 1/𝑬𝒓 𝐸𝑜𝑐 2 ∑ 1/𝑬𝒓 𝐸𝑜𝑐 2 , (6)
and the prediction of the ensemble is calculated as a weighted sum of the prediction results of all the selected sub-models, as shown in Equation ( 7), with 𝑦 ̂𝑖 and 𝑦 ̂ being the predicted value of the i-th selected sub-model and of the ensemble, respectively:

𝑦 ̂= ∑ 𝜔 𝑖 𝑦 ̂𝑖 𝐸𝑜𝐶 . ( 7
)
If none of the sub-models in the ensemble gives a local fitness that satisfies1 -𝐽 𝑆𝑖 (𝒙) < 𝜌, all the sub-models are used for calculating the prediction of the ensemble. In Equations ( 6), 𝐄𝐫 𝐸𝑜𝑐 is replaced by 𝐄𝐫 and in Equation [START_REF] Csató | Sparse on-line gaussian processes[END_REF], the weighed sum is carried out on all the sub-models.

Update of the ensemble with a new pattern (step 2.2 of Figure 1)

If the local fitness of the new data point to the FVs in each sub-model satisfies the relation 1 -𝐽 𝑆𝑖 (𝒙) > 𝜌, such data point is judged as a new FV and it is added to the first sub-model 𝑀 1 that had been trained on the training dataset (step 2.2 of Figure 1). The other sub-models are not modified with the new FV, as they represent only the patterns in the data at a certain historical period and the new FV represents the ongoing pattern of the data. A new sub-model is not created in the case of a new FV, as it enriches the ensemble without decreasing its performance on the whole data. Thus, the number of sub-models are not changed and only the sub-model 𝑀 1 is updated to follow the ongoing patterns. Once the FVs in 𝑀 1 are increased by one, the model is retrained by minimizing the MSE on the recent data points (how to choose the recent data points is explained in details in Section 3.6).

Update of the ensemble with a changed pattern (step 2.3.1 in Figure 1)

When the new data point is judged as not being a new FV, the verification of a changed FV is carried out by calculating the prediction error (absolute bias between the predicted value and the true output) of all the sub-models. If the prediction errors are all bigger than a preset threshold θ, the new data point is judged as a changed pattern and is used to replace another FV in the sub-model 𝑀 1 . Before the replacement, we need to solve two questions.

The first one is how to choose the FV in 𝑀 1 to be replaced by the new data point. The pseudo-code for Online-SVR-FID in Appendix gives a way for SVR models, based on counting the times that an FV has been a support vector in the past SVR models during the adaptive learning process and, then, the contribution in the recent SVR models are more weighted than those in the older ones. Following the same strategy, a more general way is to cumulate the contribution of the FV through a weighted sum of its value calculated in Equation ( 5) for all the data points. Suppose the contribution of each FV in 𝑀 1 is 𝑚 𝑖 : when a new data point is coming, Equation ( 5) can compute its similarity with each FV in 𝑀 1 . A bigger 𝑎 𝑖 in 𝒂 represents a larger similarity, and, thus, a bigger contribution to the prediction of the new data point: the contribution of the FV is, then, updated as 𝑚 𝑖 𝑛𝑒𝑤 = 𝛾𝑚 𝑖 + 𝑎 𝑖 , with 𝛾 a positive value smaller than one.

Once the FV in 𝑀 1 to be replaced by the new data point is selected, the second problem is how to assure that all the past patterns are stored in the ensemble. If the selected FV is unique in the ensemble, i.e. it exists only in 𝑀 1 , the replacement of this FV may cause a loss of a past pattern in the data. Thus, step 2.3.1.1 of Figure 1 proposes to "copy" the model 𝑀 1 as a new sub-model and before the replacement. The selected FV in 𝑀 1 is replaced by the new data point. With such a strategy, the changed pattern is learned by 𝑀 1 and the old pattern is not deleted from the ensemble by adding a new sub-model, which is a copy of M1 before the replacement. Note that all the submodels except 𝑀 1 are created this way and they can be seen as a copy of 𝑀 1 for t different periods. As 𝑀 1 can always follow the ongoing patterns in the data, the diversity among the submodels represents different steps of the data stream.

If the selected FV in 𝑀 1 is not unique in the ensemble, it is replaced directly by the new data point without adding a new sub-model (step 2.3.1.2).

Update of the prediction error of sub-models (step 2.4 of Figure 1)

In Section 3.3, the sub-models weights are calculated according to their prediction errors 𝐄𝐫 on the data points. After the training of the first sub-model 𝑀 1 in step 1 of Figure 1, the prediction error for 𝑀 1 is the root MSE on the whole training dataset.

When a new data point is available, part of (if the new data point is not a new FV) or all (if the new data point is a new FV) the sub-models are selected to derive the prediction of the dynamically selected ensemble, as introduced in Section 3.3. In any case, sub-model 𝑀 1 is always selected, as the online learning process assures that 𝑀 1 contains all the dimensions of the available data in RKHS while the other sub-models contain only part of it. Thus, 𝑀 1 can give a local fitness for the new data point, which is smaller than or equal to those given by other sub-models. At the end of each iteration for a new data point, the strategy for updating the prediction error of the sub-models for different situations is given below: 1) For the sub-models, except 𝑀 1 , in the dynamically selected ensemble 𝑆𝑜𝐶 for the new data point (𝒙 𝑖 , 𝑦 𝑖 ), their prediction errors are updated as 𝑬𝒓 𝐸𝑜𝐶 = 𝛽𝑬𝒓 𝐸𝑜𝐶 + |𝒚 ̂𝑖 -𝑦 𝑖 |, with the vector 𝑬𝒓 𝐸𝑜𝑐 containing their prediction errors, 𝛽 being a positive parameter smaller than one and 𝒚 ̂𝑖 the vector of predicted values of the sub-models in 𝐸𝑜𝐶. 2) For the sub-models that are not selected into 𝐸𝑜𝐶, their prediction errors are updated as 𝑬𝒓 = 𝛽𝑬𝒓 + 𝜏𝐸𝑟, with 𝐸𝑟 the maximal prediction error given by the sub-models in 𝐸𝑜𝐶 and 𝜏 a parameter bigger than one, in order to decrease the weights of these sub-models in the next iteration.

3) The case of 𝑀 1 is different from the above two types of sub-models, as it may be adaptively updated with the new data point: 3.1) If it is not updated during steps 2.2 and 2.3 of Figure 1, its prediction error is updated as step 1. 3.2) Otherwise, it is updated with the prediction error after the update, i.e. after steps 2.2 and 2.3 of Figure 1. 𝑀 1 gives a new prediction for the new data point different from the one calculated in step 2.1 of Figure 1 during the calculation of the prediction of the ensemble for the new data point. The error of the new prediction is the true error for 𝑀 1 at the end of this iteration. Its prediction error is updated with the new prediction error as 𝐸𝑟 1 = 𝛽𝐸𝑟 1 + |𝑦 ̂1,𝑛𝑒𝑤 -𝑦 𝑖 |, with 𝑦 ̂1,𝑛𝑒𝑤 being the prediction for the new data point given by the updated M1. 4) If a new sub-model is created during the online learning of the new data point, the prediction error of the new sub-model is calculated with 𝐸𝑟 𝑛+1 = 𝛽𝐸𝑟 1 + |𝑦 ̂1,𝑜𝑙𝑑 -𝑦 𝑖 |, with 𝑦 ̂1,𝑜𝑙𝑑 being the prediction for the new data point given by 𝑀 1 at step 2.1 of Figure 1, which is not updated yet with the new data point, and 𝐸𝑟 1 being the prediction error of 𝑀 1 at the beginning of this iteration in step 2.1 in Figure 1, i.e. before updating.

Retraining of the sub-model M1

Facing a new FV or a changed FV, the sub-model 𝑀 1 needs to be updated. However, it is not always possible to find a way to update directly the model, as shown in Online-SVR-FID without retraining it from scratch. In this paper, we suppose that 𝑀 1 is updated by retraining by minimizing the MSE on a number (much larger than the number of FVs in the model) of recent data points, in order to guarantee the generalization ability of the model. Suppose the last sub-model was added at the i0-th data point, when the i-th data point is coming: the objective function to minimize is the MSE on the data points from i0 to i. In order to avoid overfitting and underfitting on the recent data points, a minimal (Nmin) and a maximal (Nmax) number of the recent data points in the objective function are fixed during the retraining of 𝑀 1 , i.e. the number of the recent data points for retraining 𝑀 1 is min (max(𝑁 𝑚𝑖𝑛 , 𝑖 -𝑖 0 ) , 𝑁_𝑚𝑎𝑥).

Advantages of OE-FV

OE-FV has several advantages compared to other online learning ensemble approaches. It is an instance-based ensemble approach, which adaptively modifies the ensemble with each new data point, and, thus, can timely learn the new patterns, contrarily to data chunk-based and drift detectorbased approaches for online learning ensemble. Thus, OE-FV can instantly follow the pattern drift in the data whereas the online learning ensembles based on data chunk or sliding window can only react after a sufficient number of new data points becomes available. Also, storing all the patterns in the data makes the ensemble capable of creating new sub-models automatically when necessary, without the trouble of setting a fixed size of new data points as the data chunk-based approaches must do.

When a new sub-model needs to be created, there is no need to train this new sub-model, as it is a "copy" of the first sub-model 𝑀 1 as presented in Section 3 and the new sub-model is fixed once created. Only 𝑀 1 is updated with new data points, to follow the ongoing patterns.

The diversity of the sub-models are guaranteed, as each sub-model represents the patterns in the data during a different period, with 𝑀 1 representing the up-to-date patterns.

Finally, the new data points are all used to update the sub-models' weights, and only few of them are used to update 𝑀 1 and create new sub-models. For each new data point, instead of using all the sub-models to derive the prediction of the ensemble, only the most relevant ones are selected to form a dynamic ensemble; such strategy can reduce the computational complexity of the online learning process.

Experiments

In this section, comparisons of experimental results on several synthetic datasets and a real dataset are carried out mainly among Online-SVR-FID [START_REF] Liu | An Adaptive Online Learning Approach for Support Vector Regression[END_REF], Learn++.NSE [START_REF] Muhlbaier | An ensemble approach for incremental learning in nonstationary environments[END_REF], OWE [START_REF] Soares | An on-line weighted ensemble of regressor models to handle concept drifts[END_REF]

Prediction of synthetic dataset

The proposed online learning strategy is firstly tested on a synthetic dataset with pattern drifts and recurring patterns. The synthetic data is composed of four concept and a total of 𝑀 = 30000 data points are generated. As shown in the following equations, concepts 1 and 4 are repeated twice as recurring patterns at different period of the dataset. A Gaussian noise with a standard deviation of 0.05 is added to each input and output variable.

Concept 1: The first 1000 data points form the training dataset and the rest imitate the online learning process to be fed into the model one by one.

𝑦 𝑖 = (𝑥 𝑖 1 + 𝑥 𝑖 2 + 𝑥 𝑖 3 )/3, for 𝑖 = 1, … , 𝑀 6 , 4𝑀 6 + 1, … , 5𝑀 6 ; Concept 2
For this dataset, one more model is added for the comparisons: the SVR with sliding window for online learning, noted as SVR in Table I. The main characteristics for computation are Inter Duo i5, 2.3 GHz, and 4G RAM. for all datasets, as the maximum number of sub-models is set to be 20 and the non-pruned one does not pass this threshold.

The critical value 𝑞 𝛼 of the Bonferroni-Dunn test with 7 algorithms is 2.39 for 𝛼 = 0.10. Then, the critical difference (CD) is 1.64, from Equation 10. If the average rank difference between two methods is bigger than the CD value, their performances are significantly different.

From the average ranks in Table III, we can see that, the differences among the prediction results of OE-FV, Online-SVR-FID and OWE-Pruned for the ten synthetic datasets are not significant, while they all give significantly better results than Learn+.NSE, Learn+.NSE, OWE and SVR.

Leakage prediction in NPP

In this Section, a real case study is considered, concerning a time series dataset collected from a sensor monitoring the leak flow from the first seal of a Reactor Coolant Pump (RCP) in a NPP. The function of RCP is critical for the control and safe operation of a NPP, as it pumps cold water into the reactor to evacuate the heat produced by nuclear fission. Figure 4 is the normalized time series dataset which contains 13124 values, measured every four hours. It contains gradual, sudden and recurring data. Denoting the data as 𝑙(𝑡), the target of the work is to predict the leak flow in the next day, i.e. 𝑦(𝑡) = 𝑙(𝑡 + 6). The partial autocorrelation analysis between different time lags and the target shows that the first ten historical values are highly correlated with the target and, thus, the input vector considered is 𝒙(𝑡) = [𝑙(𝑡 -9), 𝑙(𝑡 -8), … , 𝑙(𝑡)].

After the reconstruction of the original dataset, the first 500 data points form the training dataset and the rest simulate the online learning process, which feed the ensemble one by one. The basic models are all built by SVR in this experiment.

Prediction results of the proposed ensemble approach for NPP

For the real case study, the different parameters in Section 3 are set as follows, based on grid search: ρ = 10 -6 ; θ = 0.05; γ = 0.8; β = 0.6; τ = 4; N_ min = 150; and N_ max = 500.

The online learning of a single model in [START_REF] Soares | An on-line weighted ensemble of regressor models to handle concept drifts[END_REF] and OE-FV, are firstly compared in this experiment.

In the case of updating a SVR model with Online-SVR-FID, a SVR model is trained on the training dataset and updated with the new data points, as proposed in [START_REF] Liu | An Adaptive Online Learning Approach for Support Vector Regression[END_REF]. In the experiment, there are totally 1198 new data points judged as changed patterns and 13 new data points as new patterns.

On the contrary, in the online learning ensemble with OE-FV, only 120 and 7 new data points are separately judged as changed and new patterns. OE-FV largely decreases the number of changed patterns and, thus, the computational complexity, as all the patterns are stored in the ensemble. Thus, OE-FV solves the problem that Online-SVR-FID has with recurring patterns. Figure 5 shows the prediction results of the test data points from 4600 to 6000 given by OE-FV and the positions of the changed and new patterns.

Results comparisons

In this case study, by grid search, the size 𝑁 of the data chunk in Learn++.NSE and of the time window in OWE is fixed at 500. The threshold ε for adding a new sub-model and the discounting rate in calculating the prediction error in Learn++.NSE and OWE are (0.04, 0.2) and (0.05, 0.3) respectively. In the pruned case, the maximal number of sub-models is 20. Table IV presents the MSE and Mean Absolute Relative Error (MARE), the computation time with the same computer (Inter Duo i5, 2.3 GHz, and 4G RAM) and the number of sub-models. All these approaches give comparable results considering the prediction accuracy, with Learn++.NSE giving the worst and OE-FV the best. This is caused by the update strategy integrated in the online learning ensemble. The delay during the online learning process in Learn++.NSE is longer than in OWE, and OE-FV has the shortest delay. Thus, it is verified that the instance-based approach can timely follow the ongoing patterns and give better in frequently changing environment results than data chunk-based or sliding window-based ones. The computation burden bothering the instance-based online learning ensembles is not so obvious in OE-FV. Learn++.NSE uses least time as the ensemble is updated only when a new data chunk is available. The specific strategies proposed in OE-FV, e.g. verification of new FV and changed FV, generation of new sub-models and dynamic ensemble selection, reduce the computational complexity of the online learning process and the results show that it uses much less time than OWE, which is based on the sliding window concept. The time of OE-FV is also much smaller than Online-SVR-FID, as Online-SVR-FID deletes some old patterns during the updating process and when these patterns recur, it has to relearn them before giving a good prediction result. This disadvantage increases the number of updating actions during online learning and, thus, the computational burden, and decreases the prediction accuracy. On the contrary OE-FV applies a dynamic ensemble selection strategy to select the most relevant submodels for each new data point in order to reduce the influence of the irrelevant ones. The submodels weights are updated with each new data point and the flexibility of the ensemble is increased.

In this case study, the Learn++.NSE and OWE with and without pruning give similar prediction results. As shown in [START_REF] Soares | An on-line weighted ensemble of regressor models to handle concept drifts[END_REF], a larger maximal number of sub-models does not always increase the accuracy: the accuracy is no longer improved when the number of sub-models is bigger than a certain value. Figure 6 shows the prediction results of the test data points from 5300 to 5400 given by Online-SVR-FID, Learn++.NSE, OWE and OE-FV. It is observed that OE-FV can adapt to the target faster than the others. Learn++.NSE and OWE are updated with the longest delay, as explained in the Introduction.

Fig. 6 Comparisons of the prediction results of the test data points from 5300 to 5400.

Conclusions

Based on FVS and Online-SVR-FID, a new online learning approach for SVR-based ensembles, OE-FV, is proposed. OE-FV can create an ensemble automatically from a single model. The new sub-models represent separately a certain stage of the first sub-model, whereby the diversity among them is guaranteed. To the authors' knowledge, this paradigm is used for the first time in online learning for ensembles. The dynamic ensemble selection strategy eliminates the sub-models irrelevant to the new data point, and, thus, reduces their influences on the prediction results of the ensemble. The computational burden with instance-based online learning ensemble is reduced by taking different strategies for pattern verification and a dynamic ensemble selection.

Although developed for SVR models, the online learning strategy proposed can also be used for other kernel methods.

Comparisons on two case studies show that OE-FV outperforms Online-SVR-FID and OWE in both prediction accuracy and computation time. Learn++.NSE uses less time than OE-FV, but gives slightly worse prediction results.

  Instead, FVS selects the representative data points, i.e. FVs, which are normally of a much smaller size than the training dataset, and 𝑀 1 is trained on the selected FVs through minimizing the MSE of the prediction on the whole training dataset. Such strategy can reduce the model complexity and keep the generalization ability of the model, at the same time. The process of FVS applied for selecting the FVs from the training dataset is shown in Appendix.

Fig. 3

 3 Fig. 3 Procedures for carrying out the experiments

Fig. 4

 4 Fig. 4 Data of the leak flow in RCP.

Fig. 5

 5 Fig. 5 Prediction results of OE-FV and the positions of changed and new patterns.

  1. Train the first sub-model 𝑀 1 with kernel methods on the training dataset.2. Suppose there are n sub-models (M1, M2, …, Mn) in the ensemble when a new data point is coming: 2.1 Calculate the predicted value for the new data point by a weighted-sum strategy based on the prediction errors 𝐄𝐫 of selected sub-models; 2.2 If the new data point is new FV, it is added to 𝑀 1 and then 𝑀 1 is retrained; If the existing FV to be replaced in M1 is unique in the ensemble, the model 𝑀 1 before replacement is saved as a new sub-model, named Mn+1. The selected FV in 𝑀 1 is, then, replaced by the new data point and then 𝑀 1 is retrained; 2.3.1.2 If the existing FV to be replaced in 𝑀 1 is not unique in the ensemble, no new sub-model is created and the replacement is carried out directly in 𝑀 1 and then 𝑀 1 is retrained;

	2.3 Else
	2.3.1 If the new data is a changed FV, it will be used to replace the FV that makes least contribution
	in the recent models;
	2.3.1.1

2.3.2 Else keep the sub-models in the ensemble unchanged. 2.4 Update the prediction error 𝐄𝐫 of each sub-model.

  and the proposed OE-FV, considering prediction accuracy and computation time as the performance indicators. Learn++.NSE is a typical data chunk-based approach for online learning ensemble. When a new data chunk of a fixed size N is available, a new sub-model is added if the prediction error on the new data chunk exceeds a predefined threshold ε. The sub-models' weights are updated according to their prediction error on all the data chunks, while the prediction error on the new data chunks are more weighted than those of the older ones. Learn++.NSE cannot adapt to the new patterns until a number of N new data points are available. When the ensemble is updated with the new chunk, it may not follow the ongoing patterns. There is a delay of the patterns in the ensemble compared to the patterns in the new data. And it is very difficult to decide the best size of the data chunk. In order to solve these problems with Learn++.NSE, OWE updates the sub-models weights with the prediction error when a new data point is available. The strategy for adding a new sub-model is also different from Learn++.NSE: instead of waiting for a new data chunk, a sliding window is integrated. When a new data point is available, a window of a fixed size N moves one step ahead. When the prediction error on the data points in the window exceeds a predefined threshold ε, a new model on these data points is trained. Thus, there is no need of waiting for N new data points before adding a new sub-model. It is more flexible than Learn++.NSE, but there is still a delay compared to the instance-based approaches for online learning ensemble. There is a pruned version of Learn++.NSE and OWE, which fixes a maximal number of sub-models. After the maximal number is reached, the old sub-model which gives worst prediction results on the new data points is deleted each time a new sub-model is added. Section 4.1 shows the preliminary results on several synthetic datasets and more details of the comparison are given in Section 4.2. The experimental procedures are shown in Figure3.

Table I

 I Comparisons of experimental results using Online-SVR-FID, Learn++.NSE, OWE and OE-FV.From TableI, it is observed that Learn+.NSE Pruned, OWE Pruned and OE-FV give better results than other methods, with OE-FV gives a litter better results than the other two methods. OE-FV gives better results than SVR with much less time for online learning, as, in SVR, it takes too much time to retrain the SVR model for each new data point with a fixed number of most recent data points. While in OE-FV, the ensemble is updated only when a new / changed pattern is detected. In order to compare statistically the prediction accuracy of OE-FV with the benchmark methods, ten synthetic drifting datasets (named separately Dataset 1, Dataset 2, …,Dataset 10) with 6000 data points are generated. Each dataset is divided into six data chunks of 1000 data points and the data of each chunk follows randomly one of the four concepts above. Recurring patterns (two nonconsequent chunks follow same concept) exist in the first five datasets. For each dataset, the first 500 data points form the training dataset and the rest are in the test dataset.

		Online-SVR-FID	Learn++.NSE	Learn++.NSE Pruned	OWE	OWE Pruned	OE-FV	SVR
	MSE	77*10 -4	31*10 -4	15*10 -4	48*10 -4	14*10 -4	12*10 -4	21*10 -4
	MARE	0.1557	0.0821	0.0709	0.1062	0.0700	0.0682	0.0770
	Time (s)	2613.6	18.7	20.2	352738	253.5	121.0	67090.9
	# of sub-models	1	30	20	18753	20	46	1

Table II

 II Prediction results (MSE and MARE) of different methods on ten synthetic datasets.TableIIreports the MSE and MARE on the test dataset of each synthetic drifting dataset and the bolded values are the best results given by all the methods. It is observed that OE-FV outperforms the other methods for four out of the five datasets with recurring patterns. For a dataset without recurring patterns, the results of the ensemble approaches are influenced by the sub-models trained on the past patterns. That's why Online-SVR-FID which follows always the current patterns, gives best results on such datasets. Learn+.NSE Pruned and Learn.+NSE given same prediction results

		Online-SVR-FID	Learn++.N SE	Learn++.N SE Pruned	OWE	OWE Pruned	OE-FV	SVR
					MSE
	Dataset 1	0.0065	0.0066	0.0066	0.0088	0.0050	0.0064	0.0064
	Dataset 2	0.0075	0.0092	0.0092	0.0095	0.0064	0.0057	0.0084
	Dataset 3	0.0079	0.0105	0.0105	0.0068	0.0070	0.0044	0.0096
	Dataset 4	0.0019	0.0032	0.0032	0.0022	0.0025	0.0017	0.0032
	Dataset 5	0.0029	0.0076	0.0076	0.0045	0.0048	0.0023	0.0061
	Dataset 6	0.0026	0.0099	0.0099	0.0059	0.0058	0.0066	0.0073
	Dataset 7	0.0029	0.0080	0.0080	0.0042	0.0062	0.0041	0.0076
	Dataset 8	0.0037	0.0121	0.0121	0.0073	0.0080	0.0055	0.0098
	Dataset 9	0.0025	0.0082	0.0082	0.0051	0.0049	0.0053	0.0062
	Dataset 10	0.0022	0.0067	0.0067	0.0058	0.0049	0.0044	0.0060
					MARE
	Dataset 1	0.1409	0.1208	0.1208	0.1563	0.1035	0.1194	0.1198
	Dataset 2	0.1498	0.1469	0.1469	0.1744	0.1215	0.1193	0.1434
	Dataset 3	0.1598	0.1586	0.1586	0.1517	0.1250	0.0154	0.1592
	Dataset 4	0.0543	0.0639	0.0639	0.0592	0.0523	0.0474	0.0610
	Dataset 5	0.0723	0.1215	0.1215	0.0940	0.0793	0.0713	0.0918
	Dataset 6	0.0692	0.1548	0.1548	0.1084	0.0841	0.0945	0.0982
	Dataset 7	0.0707	0.1135	0.1135	0.0829	0.0842	0.0752	0.0943
	Dataset 8	0.1004	0.2091	0.2091	0.1498	0.1283	0.1365	0.1480
	Dataset 9	0.0694	0.1381	0.1381	0.0998	0.0783	0.0910	0.0923
	Dataset 10	0.0674	0.0951	0.0951	0.0956	0.0718	0.0754	0.0843

Table III

 III Rand of different methods considering their prediction accuracy on the ten synthetic datasets.TableIIIshows the ranks of different methods considering their prediction accuracy (MSE or MARE) for the same test dataset. Rank 1 means that the corresponding method gives the highest prediction accuracy. From the average rank in the last line of TableIII, it is noted that OE-FV obtained the highest rank among all the methods. As in[START_REF] Demšar | Statistical Comparisons of Classifiers over Multiple Data Sets[END_REF], Friedman test can check if the average ranks of the different methods are significantly different from the mean rank of all the methods under the null hypothesis, and Bonferroni-Dunn test can tell if the results of OE-FV is significantly better than those of the benchmark methods. For 𝑘 methods and 𝑁 comparison results, the Friedeman statistic in Equation 8 (𝑅 𝑗 , 𝑗 = 1, … , 𝑘 For the case studied in this paper, 𝑘 = 7 methods on 𝑁 = 20 comparison results are compared as shown in TableIII. With the average rank in TableIIIfor all the method, we can calculate that 𝐹 𝐹 equals to 28.3415. 𝐹 𝐹 follows the F-distribution with 7 -1 = 6 and (7 -1) × (20 -1) = 114 degrees of freedom. The critical value of 𝐹[START_REF] Crammer | Online Passive-Aggressive Algorithms[END_REF]114) for 𝛼 = 0.05 is 2.18. As the statistic 𝐹 𝐹(28.3415) is bigger than the critical value (2.18), the null hypothesis is rejected, i.e. not all the methods perform equally for the case studies. Bonferroni-Dunn test is used to test the significance of the difference between the average rank of two methods. The difference is significant if it is equal or bigger than the critical difference (CD) (calculated in Equation[START_REF] Minku | DDD: A new ensemble approach for dealing with concept drift[END_REF] with 𝑞 𝛼 the Studentized range statistic divided by √2):

	𝜒 𝐹 2 =	12𝑁 𝑘(𝑘+1) [∑ 𝑅 𝑗 2 𝑗	-	𝑘(𝑘+1) 2 4	],	(8)
	𝐹 𝐹 =	Online-SVR-FID 𝑁(𝑘-1)-𝜒 𝐹 2 (𝑁-1)𝜒 𝐹 2 .	Learn++.N SE	Learn++.N SE Pruned	OWE	Pruned OWE	OE-FV	SVR (9)
										MSE
	Dataset 1			4				5.5	5.5	7	1	2.5	2.5
	Dataset 2			3				5.5	5.5	7	2	1	4
	Dataset 3			4				6.5	6.5	2	3	1	5
	Dataset 4			2				6	6	3	4	1	6
	Dataset 5			2				6.5	6.5	3	4	1	5
	Dataset 6			1				6.5	6.5	3	2	4	5
	Dataset 7			1				6.5	6.5	3	4	2	5
	Dataset 8			1				6.5	6.5	3	4	2	5
	Dataset 9			1				6.5	6.5	2	3	4	5
	Dataset 10 𝐶𝐷 = 𝑞 𝛼 √	𝑘(𝑘+1) 6𝑁 .	1				6.5	6.5	4 MARE	3	2	5
	Dataset 1			6				4.5	4.5	7	1	2	3
	Dataset 2			6				4.5	4.5	7	2	1	3
	Dataset 3			6				3.5	3.5	7	2	1	5
	Dataset 4			3				6.5	6.5	4	2	1	5
	Dataset 5			2				6.5	6.5	5	3	1	4
	Dataset 6			1				6.5	6.5	5	2	3	4
	Dataset 7			1				6.5	6.5	3	4	2	5
	Dataset 8			1				6.5	6.5	5	2	3	4
	Dataset 9			1				6.5	6.5	5	2	3	4
	Dataset 10			1				5.5	5.5	7	2	3	4
	Average Rank	2.400		5.975	5.975	4.600	2.600	2.025	4.425
	is the mean rank of method j) follows a 𝜒 𝐹 2 distribution with 𝑘 -1 degrees of freedom and the
	statistic in Equation 9 follows the F-distribution with 𝑘 -1 and (𝑘 -1)(𝑁 -1) degrees of
	freedom,

Table IV

 IV Comparisons of experimental results using Online-SVR-FID, Learn++.NSE, OWE and OE-FV.

		Online-SVR-FID	Learn++.NSE	Learn++.NSE Pruned	OWE	OWE Pruned	OE-FV
	MSE	13*10 -4	16*10 -4	16*10 -4	12*10 -4	12*10 -4	8.6*10 -4
	MARE	0.0977	0.1009	0.1009	0.0879	0.0882	0.0761
	Time (s)	460.117	8.3607	8.0682	30485	188.394	51.299
	# of sub-models	1	26	20	7513	20	13

The proposed approach for online learning ensemble: OE-FVAs mentioned in previous sections, the main objective of this ensemble is to store all the past patterns in the data, propose strategies to build automatically new sub-models, update their weights and decrease the computational burden for instance-based approaches for online learning ensemble. The whole idea is based on the FVS in[START_REF] Baudat | Feature vector selection and projection using kernels[END_REF] and Online-SVR-FID in[START_REF] Liu | An Adaptive Online Learning Approach for Support Vector Regression[END_REF]. OE-FV builds automatically an ensemble from the first sub-model trained on the training dataset. All the sub-models are expected to represent the characteristics of the data during a certain period and once the old patterns recur, the most relevant sub-models are selected by FVS to derive the prediction. As FVS is developed for the kernel methods, OE-FV can be applied for all kernel-based ensembles, e.g. sub-models trained with kernel ridge regression[START_REF] Hoerl | Ridge Regression: Biased Estimation for Nonorthogonal Problems[END_REF], SVR[START_REF] Smola | A Tutorial on Support Vector Regression[END_REF], Gaussian process[START_REF] Seeger | Gaussian processes for machine learning[END_REF] etc. The main procedure is shown in Figures1 and 2. OE-FV builds an ensemble sequentially from the first sub-model, named 𝑀 1 , that is trained on the preliminary training dataset. All the other submodels can be seen as an updated "copy" of 𝑀 1 at one instance during the data stream developing process. These sub-models are expected to be different from each other and represent the data at a certain period of the stream. Only the sub-model 𝑀 1 is adaptively updated with new data points, while the other sub-models are fixed once created and stored.

Acknowledgement

The authors want to thank the author Symone Gomes Soares of [22] who shared the original code for the online learning ensemble in her paper.

Appendix

Online-SVR-FID is divided into two parts, i.e. Offline Training and Online Learning. The Offline Training contains the feature vector selection and SVR model training, while the Online Learning contains the actions to take, when facing a new or a changed pattern.

The pseudo-code of Online-SVR-FID is repeated below. For i = model is more weighted than that in the old models; otherwise it is kept unchanged. 5. When a change is detected with respect to the old patterns, the first step is to calculate the values 𝒂 for the new data point according to Equation [START_REF] Lee | An incremental adaptive neural network model for online noisy data regression and its application to compartment fire studies[END_REF]. Then, among all the FVs in the model with non-zero values in 𝒂, the one with least contribution, say 𝑚 𝐼 , is deleted from the model using Decremental Learning as in [START_REF] Ko | From dynamic classifier selection to dynamic ensemble selection[END_REF] and 𝑚 𝐼 is reset to zero. If there are several FVs with the same contribution and the least contribution, the FV to be replaced is selected as the oldest one among them. 6. The new data point is added to the model using Incremental Learning in [START_REF] Ko | From dynamic classifier selection to dynamic ensemble selection[END_REF] and it inherits the contribution 𝑚 𝐼 , which is zero for now. The vector 𝒎 and the feature space S are updated, and also the contribution of the FV is updated according to the rules in step 4 above.


Ben-Michael Kohli

THE LINKS-GOULD INVARIANT AS A CLASSICAL GENERALIZATION OF THE ALEXANDER POLYNOMIAL ?

Keywords: 2010 Mathematics Subject Classication. 57M27 Link, knot, Alexander-Conway polynomial, Links-Gould invariant, genus, beredness

In this paper we conjecture that the Links-Gould invariant of links, that we know is a generalization of the Alexander-Conway polynomial, shares some of its classical features. In particular it seems to give a lower bound for the genus of links and to provide a criterion for beredness of knots. We give some evidence for these two assumptions.

LG(L; t 0 , -t

 LG(L; t 0 , t

Knowing this, it is natural to wonder :

Question 0.1. Are there properties of ∆ that extend to LG ?

, it is in essence a classical invariant derived from a presentation matrix of the rst homology group of the innite cyclic covering of the complement of a given link in S 3 [1]. Therefore we can ask if some of ∆'s homological properties extend to LG.

In that spirit, let us recall that a conjecture by Ishii [12] states :

Conjecture 0.2. The LG polynomial LG(K; t 0 , t 1 ) = i,j a ij t i 0 t j 1 of an alternating knot K is "alternating", that is : a ij a kl 0 if i + j + k + l is even, and a ij a kl 0 otherwise.

Introduction

The Links-Gould invariant LG(L; t 0 , t 1 ) is a two variable polynomial quantum invariant. It is derived from a one parameter family of representations of quantum superalgebra U q (gl(2|1)) [START_REF] Wit | On the Links-Gould invariant of links[END_REF][START_REF] Links | Two variable link polynomials from quantum supergroups[END_REF]. It is part of a larger family of Links-Gould invariants LG m,n , m, n ∈ N * [START_REF] Wit | An innite suite of Links-Gould invariants[END_REF].

It is worth noticing that the Alexander-Conway polynomial of a link ∆ L can be recovered from LG in at least two ways. David de Wit, Atsushi Ishii and Jon Links showed [START_REF] Wit | Innitely many two-variable generalisations of the Alexander-Conway polynomial[END_REF] Though if you think about it this is not a straightforward generalization of the well known similar property for the Alexander polynomial [START_REF] Murasugi | On the genus of the alternating knot II[END_REF][START_REF] Crowell | Genus of Alternating Link Types[END_REF] for either of the two evaluations we know t 0 t 1 = 1 and t 0 t 1 = -1, it still can be thought of as the trace of a similar behavior.

In the following we give evidence for more positive answers to Question 0.1. We conjecture that the span of the LG invariant is a lower bound for the genus of a link. Conjecture 0.3. Set L a link in S 3 and µ the number of its components.

• I: span(LG(L; t 0 , t 1 )) 2(2g(L) + µ -1),

• II: If L is alternating, then inequality I is an equality.

We also conjecture that for bered knots, there are conditions on the leading coecients of the LG polynomial.

Conjecture 0.4. Set K a knot in S 3 .

• I: If K is bered then LG(K) is monic,

• II: If K is alternating, the converse is true as well.

We base these conjectures on computations for the rst prime knots and on partial skein relations for LG that allow its evaluation on various innite families of links. Notice that if the genus conjecture were true, LG would systematically give a better lower bound for the genus of a link than the one given by the Alexander invariant. Also, the criterion we conjecture for bered knots would rene the well known similar statement for ∆.

A proof of these two statements would show quantum invariant LG can be used to nd information on the geometry of links.

1. The Links-Gould invariant and the genus of links 1.1. Denitions. We recall the denition of the Links-Gould invariant of links. We then give the denition of the genus of a link and recall the connection between the genus of a link and its Alexander invariant. Denition 1.1. Set K := C(t ± 1 2 0 , t ± 1 2 1 ). Let W =< e 1 , . . . , e 4 > be a four-dimensional Kvector space. The following linear map R, expressed in basis (e 1 ⊗ e 1 , e 1 ⊗ e 2 , e 1 ⊗ e 3 , e 1 ⊗ e 4 , e 2 ⊗ e 1 , e 2 ⊗ e 2 , e 2 ⊗ e 3 , . . .), is an automorphism of W ⊗ W and an R-matrix ( [START_REF] Wit | On the Links-Gould invariant of links[END_REF], p.186) : [START_REF] Kohli | On the Links-Gould invariant and the square of the Alexander polynomial[END_REF], R is multiplied by -1. We chose this convention here for it is the one Ishii uses in [START_REF] Ishii | The Links-Gould polynomial as a generalization of the Alexander-Conway polynomial[END_REF][START_REF] Ishii | Algebraic links and skein relations of the Links-Gould invariant[END_REF], two papers that are interesting for our study.

                               
Theorem 1.3. Let L be an oriented link, and b ∈ B n a braid with closure L. Set µ the linear map dened by

µ =     t -1 0 . . . . -t 1 . . . . -t -1 0 . . . . t 1     ∈ End(W ).
Then :

1) there exists an element c ∈ K such that trace 2,3,...,n ((id [START_REF] Cha | KnotInfo: Table of Knot Invariants[END_REF] c is an oriented link invariant called Links-Gould invariant of L. We will denote it by

W ⊗ µ ⊗n-1 ) • b n R (b)) = c.id W ,
LG(L; t 0 , t 1 ).

Remark 1.4. In fact LG(L; t 0 , t 1 ) ∈ Z[t ±1 0 , t ±1 1 ] [12]. Remark 1.5.
With the notations we use, LG(L; q -2α , q 2α+2 ) is the Links-Gould invariant introduced in [START_REF] Wit | On the Links-Gould invariant of links[END_REF], using a one parameter family of representations of quantum superalgebra

U q (gl(2|1)). Denition 1.6. (Seifert surface for a link) Set L a link in S 3 . A Seifert surface for L is a compact, connected, orientable surface Σ ⊂ S 3 such that ∂Σ = L.
Such a surface exists for any link according to Seifert's algorithm [START_REF] Seifert | Über das Geschlecht von Knoten[END_REF].

Remark 1.7. Any Seifert surface Σ being connected and orientable, one can dene the genus g(Σ) of Σ :

χ(Σ) = 2 -2g(Σ) -µ
where χ(Σ) is the Euler characteristic of Σ and µ is the number of components of link L. Denition 1.8. (genus of a link) Let L be a link in S 3 . The genus g(L) of L is g(L) := min{g(Σ), Σ Seifert surface for L}. Denition 1.9. (Alexander polynomial of a link) Set L a link in S 3 with µ components and choose Σ a Seifert surface for L. Then H 1 (Σ, Z) is a free abelian group of rank Z) with the pusho of the j th generator, then V = (v ij ) is a Seifert matrix for L. The Alexander polynomial is computed from such a Seifert matrix setting

1 -χ(Σ) = 2g(Σ) + µ -1. If v ij is the linking number in S 3 of the i th generator of H 1 (Σ,
∆ L (t) = det(tV -t V ) ∈ Z[t, t -1 ].
With this denition, ∆ L is determined up to multiplication by ±t n , n ∈ Z. The standard Alexander normalization consists in picking the representative with positive constant term. The Alexander-Conway normalization corresponds, at least in the case of a knot K, to choosing the symmetric Laurent polynomial with ∆ K (1) = 1.

Proposition 1.10. For any link L, deg(∆ L (t)) 2g(L) + µ -1.

So the degree of ∆ L gives a lower bound on the genus of link L.

1.2. The genus conjecture. We believe that Proposition 1.10 can be extended to the similar statement expressed in Conjecture 0.3. We will explain how and why it would be an extension of 1.10. The goal of section 2 is to give a range of evidence to support that conjecture. Denition 1.11.

Set P ∈ Z[t ±1 0 , t ±1 1 ]. For (n, m) ∈ Z 2 , we dene deg(t n 0 t m 1 ) := n -m.
For a general P = i,j∈Z a ij t i 0 t j 1 , we can extend that denition to introduce the span of P :

span(P ) := max{deg(t i 0 t j 1 ), (i, j) ∈ Z 2 | a ij = 0} -min{deg(t i 0 t j 1 ), (i, j) ∈ Z 2 | a ij = 0}. Remark 1.
12. The span satises the usual elementary degree properties : span(P Q) = span(P ) + span(Q), span(P + Q) max{span(P ), span(Q)} if P and Q are symmetric Laurent polynomials. Conjecture 0.3 generalizes Proposition 1.10 since this well known result shows Conjecture 0.3 is true when t 0 t 1 = 1 and t 0 t 1 = -1 via the evaluations we already mentionned

LG(L; t 0 , -t -1 0 ) = ∆ L (t 2 0 ) ; LG(L; t 0 , t -1 0 ) = ∆ L (t 0 ) 2 .
These evaluations also explain why our denition for the span was natural to try and push the lower bound a little further. Proof. Since LG(L; t 0 , -t -1

0 ) = ∆ L (t 2 0 ), 2deg(∆ L (t)) = deg(∆ L (t 2 )) = span(LG(L; t, -t -1 )).
So to prove 2, we wish to show span(LG(L; t, -t -1 )) span(LG(L; t 0 , t 1 )).

If we denote

LG(L; t 0 , t

1 ) = i,j∈Z/ a ij =0 a ij t i 0 t j 1 , then LG(L; t, -t -1 ) = k∈Z ( i,j∈Z/ a ij =0 i-j=k a ij (-1) j )t k .
This clearly shows that if the coecient in front of t k in LG(L; t, -t -1 ) is non zero, then there is at least one non zero coecient in front of a monomial of degree k in the expression of LG(L; t 0 , t 1 ), which yields 2. Moreover, some examples where the equality does not hold are given in Proposition 2.1. Now suppose I holds for any link and set L an alternating link. Then [START_REF] Crowell | Genus of Alternating Link Types[END_REF], Theorem 3.5,

states deg(∆ L (t)) = 2g(L) + µ -1.
So we have the following inequality chain :

2.(2g(L) + µ -1) span(LG(L; t 0 , t 1 ))

(Conjecture 0.3) 2deg(∆ L (t)) (point 2) = 2.(2g(L) + µ -1)
(reference [START_REF] Crowell | Genus of Alternating Link Types[END_REF]) CQFD

Evidence supporting the genus conjecture

We wish to give evidence of the likeliness of Conjecture 0.3. In particular we verify the bound for small prime knots, prove it for several innite families of knots and links and verify that the genus conjecture holds on an untwisted Whitehead double of the trefoil knot, which is a counter example due to Hugh Morton in a ressembling situation we will explain.

2.1. Less than 13 crossing prime knots. When we consider knots, µ is equal to 1 and the inequality becomes span(LG(K; t 0 , t 1 )) 4g(K).

We tested that inequality on all prime knots with less than 12 crossings, and on a large selection of non alternating prime knots with 13 crossings. To do that, we used the computations of LG for prime knots one can access via David de Wit's LINKS-GOULD EXPLORER [START_REF] Wit | Links-Gould Explorer[END_REF]. To nd genus information up to 12 crossings, we used Cha and Livingston's KNOT-INFO [START_REF] Cha | KnotInfo: Table of Knot Invariants[END_REF]. For non alternating 13 crossing prime knots, data is obtained from Stoimenow's website KNOT DATA TABLES [START_REF] Stoimenow | Knot Data Tables[END_REF]. Knots are listed with respect to the HTW ordering for tables of prime knots of up to 16 crossings [START_REF] Hoste | The rst 1,701,936 knots[END_REF].

The reason why we did not test all non alternating 13 crossing prime knots is explained in [START_REF] De Wit | Where the Links-Gould invariant rst fails to distinguish nonmutant prime knots[END_REF] : The LINKS-GOULD EXPLORER's database contains evaluations only for LG of knots with string index at most 5, and from time to time 6 or 7. Indeed, the memory required increases dramatically with braid width. This still provides values for LG for 2096 non alternating prime knots with 13 crossings among the 5110 which exist. Proposition 2.1.

1 : Conjecture 0.3 holds for every knot tested. In particular, for all alternating knots tested, the equality holds. 2 : For all prime knots with less than 10 crossings, the span of LG exactly is 4 times the genus of the knot. 3 : The list of prime knots with 11 or 12 crossings where there is no equality is the following : Once it is proved to be true, the span inequality is meaningful when 2deg(∆ L (t)) < span(LG(L; t 0 , t 1 )). However, as long as it remains a conjecture, the hard case for the inequality is when deg(∆ L (t)) = 2g(L) + µ -1 precisely because there is no choice on the value of the span of LG for it not to be a counter-example.

2.2. The untwisted Whitehead double of the trefoil knot. Here we compute the Links-Gould polynomial of the untwisted double of the trefoil knot that is drawn is Figure 1. This is an interesting knot to study since it is a counterexample to a genus type bound for another generalization of the Alexander-Conway polynomial : the HOMFLY-PT polynomial. Precisely, Hugh Morton shows in [START_REF] Morton | Seifert circles and knot polynomials[END_REF], theorem 2, that the monomial of highest degree with respect to the Alexander variable in the HOMFLY-PT invariant gives a lower bound for 2g(L) + µ -1, where g(L) is the canonical genus of link L. The double of the trefoil is the example Morton gives to show that degree is not in general a lower bound for 2g(L) + µ -1. There is no such problem here : Proposition 2.3. The untwisted Whitehead double of the right handed trefoil with positive clasp K 0 has the following properties :

• g(K 0 ) = 1, • ∆ K 0 (t) = 1,
• A braid presentation of K 0 in braid group B 6 as a word in the standard Artin generators σ 1 , . . . , σ 5 is :

σ 4 σ 3 σ 2 σ -1 3 σ -1 4 σ 5 σ 3 σ 2 σ 1 σ -1 2 σ -1 3 σ 5 σ 4 σ 3 σ -1 4 σ -1 5 σ 3 σ 2 σ -1 3 σ 2 σ 2 1 σ -1 2 , • LG(K 0 ; t 0 , t 1 ) = 3 -4t 1 + 2t 2 1 -4t 0 + 6t 1 t 0 -2t 2 1 t 0 + 2t 2 0 -2t 1 t 2 0 -2t 2 1 t 2 0 + 4t 3 1 t 2 0 - 2t 4 1 t 2 0 + 4t 2 1 t 3 0 -10t 3 1 t 3 0 + 8t 4 1 t 3 0 -2t 5 1 t 3 0 -2t 2 1 t 4 0 + 8t 3 1 t 4 0 -8t 4 1 t 4 0 + 2t 5 1 t 4 0 -2t 3 1 t 5 0 + 2t 4 1 t 5 0 + 4t 5 1 t 5 0 -6t 6 1 t 5 0 + 2t 7 1 t 5 0 -6t 5 1 t 6 0 + 8t 6 1 t 6 0 -2t 7 1 t 6 0 + 2t 5 1 t 7 0 -2t 6 1 t 7
0 . So the span of LG(K 0 ; t 0 , t 1 ) is 4 and the span inequality is veried in this case. Remark 2.4. The value of LG(K 0 ; t 0 , t 1 ) was obtained by direct computation of the formula given by Theorem 1.3 with the R-matrix in Denition 1.1 using MATHEMATICA 10.

2.3. Innite families of knots where Conjecture 0.3 can be veried thanks to partial skein relations we know. Here we verify the genus bound on several innite families of knots or links. To do that, we will use basic Alexander-type properties of LG we will recall, and partial skein relations that will make the computations practicable.

Some properties of

LG and useful skein relations. To compute LG for innite families of knots, we need to have a more ecient way to evaluate it than simply using the formula in 1.3. We rst recall some general facts about the LG polynomial.

Proposition 2.5. The Links-Gould polynomial satises the following properties :

• LG( ) = 1,
• Denoting L * the reexion of L, LG(L * ; t 0 , t 1 ) = LG(L; t -1 0 , t -1 1 ),

• We have the following symmetry : LG(L; t 0 , t 1 ) = LG(L; t 1 , t 0 ). Indeed LG does not detect inversion, • For L and L two links, denoting L#L their connected sum : LG(L#L ) =

LG(L)LG(L ),

• If L = L L is the split union of L and L , then LG(L) = 0. Proof. For proofs of these facts, we refer the reader to [START_REF] Ishii | Algebraic links and skein relations of the Links-Gould invariant[END_REF][START_REF] Wit | On the Links-Gould invariant of links[END_REF][START_REF] Wit | An innite suite of Links-Gould invariants[END_REF]. CQFD Remark 2.6. The last two points show that LG and ∆ behave similarily concerning sums and disjoint unions.

Let us also cite a list of skein relations that are known to be true for LG. Whether the associated skein module is generated by the unknot or not is a problem pointed out by Ishii [START_REF] Ishii | Algebraic links and skein relations of the Links-Gould invariant[END_REF]. It is to the best of our knowledge an open question. Proposition 2.7. LG veries the following skein relations.

Skein relation [START_REF] Alexander | Topological invariants of knots and links[END_REF] :

LG

+ (1 -t 0 -t 1 )
LG

+(t 0 t 1 -t 0 -t 1 )LG + t 0 t 1 LG = 0.

Skein relation (2) :

LG

+ (1 -t 0 -t 1 )
LG

+(t 0 t 1 -t 0 -t 1 )LG + t 0 t 1 LG = 0.

Skein relation (3) :

LG

+ (t 0 t 1 -t 0 -t 1 + 2)
LG

-(t 0 t 1 -t 0 -t 1 + 2)LG -LG = 0
.

Skein relation (4) :

LG -(t 0 t 1 + 1)

LG

+t 0 t 1 LG + 2(t 0 -1)(t 1 -1)LG = 0.
Proof. See [START_REF] Ishii | Algebraic links and skein relations of the Links-Gould invariant[END_REF][START_REF] Wit | On the Links-Gould invariant of links[END_REF][START_REF] Links | Two variable link polynomials from quantum supergroups[END_REF]. CQFD Remark 2.8. ( 1) and ( 2) are equivalent, by adding each time a well chosen tangle from the left.

Remark 2.9. As explained in [START_REF] Ishii | Algebraic links and skein relations of the Links-Gould invariant[END_REF], (4) is a consequence of ( 2) and (3). Remark 2.10. Set V the 4-dimensional irreducible U q (gl(2|1))-module that gives rise to the Links-Gould invariant. Then the tensor product of two copies of V decomposes with respect to the U q (gl(2|1))-module structure.

V ⊗ V = V 1 ⊕ V 2 ⊕ W , with dimV 1 , dimV 2 = 4 and dimW = 8. Morevover, V 1 , V 2
and W are non isomorphic irreducible U q (gl(2|1))-modules. For details, see [START_REF] Geer | Multivariable link invariants arising from sl(2|1) and the Alexander polynomial[END_REF][START_REF] Wit | On the Links-Gould invariant of links[END_REF]. Using this and denoting A := U q (gl(2|1)), we have the following identication :

End A (V ⊗ V ) End A (V 1 ) ⊕ End A (V 2 ) ⊕ End A (W ) C(t ±1 0 , t ±1 1 ) 3 .
In particular, for any three (2, 2)-tangles such that the associated maps in End A (V ⊗ V ) are linearly independent, any other can be expressed as a linear combination of the rst three. This potentially generates a great variety of skein relations for LG.

Remark 2.11. Using points 2 and 3 of Proposition 2.5, we can modify the previous skein relations : orientation of the strands, signs of the crossings. We will use these modied relations, though we will not write them down here. Corollary 2.12. Using notations in [START_REF] Ishii | Algebraic links and skein relations of the Links-Gould invariant[END_REF],

LG satises the following skein relations :

• LG n half twists = (-1) n (t 0 +1)(t 1 +1) + t n 0 (t 0 +1)(t 0 -t 1 ) + t n 1 (t 1 +1)(t 1 -t 0 )
LG -

(-1) n (t 0 +t 1 ) (t 0 +1)(t 1 +1) + t n 0 (t 1 -1) (t 0 +1)(t 0 -t 1 ) + t n 1 (t 0 -1) (t 1 +1)(t 1 -t 0 )
LG

+ (-1) n t 0 t 1 (t 0 +1)(t 1 +1) - t n 0 t 1 (t 0 +1)(t 0 -t 1 ) - t n 1 t 0 (t 1 +1)(t 1 -t 0 )
LG .

•

LG n full twists

= t n 0 t n 1 -1 t 0 t 1 -1 LG + 1 - t n 0 t n 1 -1 t 0 t 1 -1 LG + 2(t 0 -1)(t 1 -1) t 0 t 1 -1 n - t n 0 t n 1 -1 t 0 t 1 -1 LG . • LG n full twists = a 1 (n)LG + a 2 (n)LG + a 3 (n)LG
, where :

a 1 (n) = t n 0 t n 1 -1 t 0 t 1 -1 , a 2 (n) = 2n(t 0 -1)(t 1 -1) t 0 t 1 -1 -a 1 (n) (t 0 t 1 +1)(t 0 -1)(t 1 -1) t 0 t 1 -1 + 1 , a 3 (n) = (t 0 -1)(t 1 -1)a 1 (n) + 1.
We will now use all these properties to compute LG, or at least its span, on some innite families of links.

2-bridge links.

A 2-bridge link is a link with bridge number 2. As explained in [START_REF] Kanenobu | Genus and Kauman polynomial of a 2-bridge knot[END_REF], an oriented 2-bridge link can always be written in terms of the two generators S 1 and S 2 of 3-string braid group B 3 . We use the notations one can nd in [START_REF] Kanenobu | Genus and Kauman polynomial of a 2-bridge knot[END_REF][START_REF] Ishii | Algebraic links and skein relations of the Links-Gould invariant[END_REF], setting D(b 1 , b 2 , . . . , b m ) the oriented 2-bridge link drawn in Figures 3 and4. We can suppose b 1 , . . . , b m > 0, thereby choosing an alternating diagram to represent any 2-bridge link. In particular, any 2-bridge link is alternating, so the inequality in Conjecture 0.3 should be an equality. where µ is the number of components [START_REF] Siebenmann | Exercices sur les noeuds rationnels[END_REF]. Proof. First we note that a 1 (n), a 2 (n), a 3 (n) are symmetric polynomials with respect to variables t 0 and t 1 . We can compute the span in each case.

span(a 1 (n)) = 0, span(a 2 (n)) = 2, span(a 3 (n)) = 2.
We will indicate by ãi (n) the quantity a i (n)(t -1 0 , t -1 1 ). Let's prove the span equality by induction on m. m = 1 Using the mirror of skein relation 3 of Corollary 2.12 :

LG LG

(D(b 1 , b 2 )) = a 1 (b 2 )LG(D(b 1 -1)) + a 2 (b 2 )LG(D(b 1 )) + a 3 (b 2 )LG( ).
The second part of the sum has span 2 + 2 = 4. The third part has span 2 + 0 = 2.

More care has to be taken with the rst term, and in particular with

LG(D(b 1 -1)). If b 1 -1 > 0,
LG(D(b 1 -1)) has span 2. In the other case, D(0) = so LG(D(0)) = 0. So in any case the span of the sum is 4.

Let us now set m 3 and suppose the equality stands for any D(b 1 , . . . , b k ) with k m -1. For D(b 1 , . . . , b m ) we can apply skein relation 3 of 2.12 or its mirror image to the crossings that correspond to S -2bm 1 or S 2bm 2 depending on whether m is odd or even. For instance K 0 is the unknot, K 1 is the trefoil knot and K 2 is knot 5 2 . Proposition 2.17. For n = 0, g(K n ) = 1. Proposition 2.18.

• LG(K 0 ) = LG( ) = 1. • For n 1, LG(K n ) = (-t -2 0 t -1 1 -t -1 0 t -2 1 + t -2 0 + 2t -1 0 t -1 1 + t -2 1 -t -1 0 -t -1 1 + 1)(t -1 0 t -1 1 ã1 (n -1) + 1) + (t -1 0 -1) 2 (t -1 1 -1) 2 (t -1 0 t -1 1 + 1) ã1 (n -1) -2(n -1) t -1 0 t -1 1 -1 + ã1 (n -1)(t -1 0 -1) 2 (t -1 1 -1) 2 -t -1 0 t -1 1 ã1 (n -1). p q r Figure 6.
It has span 4.

• For n -1, LG(K n ) = (t -1 0 -1)(t -1 1 -1) a 1 (-n) + (t 0 -1)(t 1 -1) 2n + a 1 (-n)(t 0 t 1 + 1) t 0 t 1 -1 + (t 0 -1)(t 1 -1)a 1 (-n) + 1.
It has span 4 as well. Proof. When n 1, we rst write the third skein relation of 2.12 for n -1 full twists. On each of the three links that appear, we use the rst point of 2.12. We nd the formula written in the theorem. A close look at that expression shows that

span(LG(K n )) 4.
To see it is equal to 4, we can for example evaluate t 1 = -t -1 0 . We know we will nd (and can verify)

LG(K n )(t 0 , -t -1 0 ) = ∆ Kn (t 2 0 ) = nt 2 0 -(2n -1) + nt -2 0 .
So span(LG(K n )) 4. Similar computations can be made when n -1. CQFD 2.3.4. Pretzel knots. Denition 2.19. Set p, q, r ∈ Z. The (p, q, r)-pretzel link L(p, q, r) is a union of three pairs of strands half-twisted p, q, r times and attached along the tops and bottoms as shown in Figure 6. The half-twists are oriented according to whether the integer is positive or negative.

For example, pretzel knot L(-2, 3, 7) is represented in Figure 7.

Proposition 2.20. L(p, q, r) is a knot ⇐⇒ at most one of the three integers p, q and r is even . In that case pretzel knot L(p, q, r) is denoted by K(p, q, r).

In [START_REF] Kim | Some invariants of pretzel links[END_REF], Kim and Lee explicit the genus for all pretzel knots. Verifying the genus conjecture on this family of knots is quite interesting since the genus does not behave the same way as a function of parameters (p, q, r) in all cases. The next theorem is proved in [START_REF] Kim | Some invariants of pretzel links[END_REF]. Theorem 2.21. Let p, q, r be integers. The genus of K(p, q, r) is as follows :

1 : K(p, ±1, ∓1), K(±2, ∓1, ±3) have genus 0 for any p, r -2 half twists smaller than 4.

3 For r = 1, 3, K(2, -1, r) is the unknot. If r 5, K(2, -1, r) is drawn in Figure 8 once it is simplied. The same kind of isotopy can be operated on K(2, -1, r) when r -1 and the result is shown in Figure 9.

For example if r 5 we can apply skein relation 1 of Corollary 2.12 to the r -2 half twists.

LG

(K(2, -1, r)) = -1 (t 0 + 1)(t 1 + 1) + t r-2 0 (t 0 + 1)(t 0 -t 1 ) + t r-2 1 (t 1 + 1)(t 1 -t 0 ) LG - -t 0 -t 1 (t 0 + 1)(t 1 + 1) + t r-2 0 (t 1 -1) (t 0 + 1)(t 0 -t 1 ) + t r-2 1 (t 0 -1) (t 1 + 1)(t 1 -t 0 )
LG( )

+ -t 0 t 1 (t 0 + 1)(t 1 + 1) - t r-2 0 t 1 (t 0 + 1)(t 0 -t 1 ) - t r-2 1 t 0 (t 1 + 1)(t 1 -t 0 )
LG( )

= (t 0 -t 1 )(t 0 t 1 + 1) -t r-1 0 (t 1 -1)(t 1 + 1) + t r-1 1 (t 0 -1)(t 0 + 1) (t 0 + 1)(t 1 + 1)(t 0 -t 1 )
The numerator has span 2r -2 and the denominator has span 4. So

span(LG(K(2, -1, r))) = (2r -2) -4 = 2r -6 = 4 r -3 2 = 4 |r -2| -1 2 .
Computations can be led in a similar way in the other case.

4 If p is positive and even, and q, r are positive and odd. Choosing an orientation for K(p, q, r), we can apply skein relation 2 of 2.12 on the p half twists and skein relation 1 of 2.12 on the q and r half twists. As in a previous case, we get a sum of 27 terms, each of which can be computed easily. All these terms have a span smaller than 2q + 2r = 4g. Therefore we have the inequality in this case. If p is positive, even and dierent from 2, q = -1, and r is odd and negative. Choosing an orientation here again, we can use skein relation 2 on the p half twists and skein relation 1 on the r half twists. Each of the 9 parts of the sum such obtained has a span smaller than 2 -2r. So once again

span(LG(K(p, -1, r))) 2 -2r = 4 1 -r 2 = 4g.
If p is even and positive, q is odd negative and dierent from -1, and r is odd and negative. An extended computation similar to the two previous ones proves the bound in this case as well.

5 If p is even, positive and dierent from 2, q = -1, and r is positive, odd and dierent from 1. Applying skein 2 of 2.12 on the p/2 full twists, we nd three links, each of which is the closure of a power of generator σ 1 of B 2 . We can then use skein relation 1 of 2.12 on each of these links to nd that these three terms have a span smaller than 2r -2 = 4 r-1 2 . If p is even and positive, q is odd and positive, r is odd and negative, and (q, r) = (1, -1). This is the most tricky case. Indeed, if we compute LG naively using skein relations 1 and 2 as we did for the moment, some parts of the sum we obtain have a span larger than 4g = 2q -2r -4. We therefore have to look at these particular terms to see that what goes past the bound we hope actually compensates. This is achieved in the appendix (section 4).

If p is positive and even, q is odd and q -3, and r is odd and positive. K(p, q, r) can be isotoped as follows :

K(p, q, r) = K(r, q, p) = K(p, r, q). The last form K(p, r, q) shows that this case is a consequence of the two previous cases of point [START_REF] Wit | On the Links-Gould invariant of links[END_REF]. The Links-Gould we dened and studied up to now is a particular case of a larger family of Links-Gould invariants, introduced by David de Wit in [START_REF] Wit | An innite suite of Links-Gould invariants[END_REF]. We will write LG m,n where m, n are positive integers. Each invariant is derived from a highest weight representation of U q (gl(m|n)). The invariant we denoted LG is LG 2,1 .

In [START_REF] Wit | Innitely many two-variable generalisations of the Alexander-Conway polynomial[END_REF], D. De Wit, A. Ishii and J. Links proved that

LG n,1 (L; t 0 , e -2iπ/n t -1 0 ) = ∆ L (t n 0 ) for any n ∈ N * . They also conjectured that for any n ∈ N *

LG n,1 (L; t 0 , t -1 0 ) = ∆ L (t 0 ) n .

This was proved by the author when n = 2, 3 by studying the link between the R-matrix representations of braid groups that give birth to the Links-Gould invariants and the Burau representations of B n [START_REF] Kohli | On the Links-Gould invariant and the square of the Alexander polynomial[END_REF]. More recently, Bertrand Patureau-Mirand and the author proved the statement for any n by showing the -1 specialization of quantum supergroup U q (gl(n|1)) shares properties with super Hopf algebra U q (gl(1|1)) [START_REF] Kohli | Other quantum relatives of the Alexander polynomial through the Links-Gould invariant[END_REF]. Though we lack computations for LG m,n when (m, n) = (1, 1), (2, 1), the previous evaluations extend those we have for LG 2,1 so a potential homological interpretation for LG 2,1 should extend to

LG n,1 as well.

Question 2.24. Set L a link in S 3 and n 3. Do we have, as it seems to be the case when n = 2 :

• I: span(LG n,1 (L; t 0 , t 1 )) n(2g(L) + µ -1),

• II: If L is alternating, then inequality I is an equality ? Remark 2.25. For example, the equality holds for all prime knots with less than 10 crossings when n = 3.

Remark 2.26. As a consequence, one could ask, as n tends to innity, if span(LG n,1 (L; t 0 , t 1 )) ∼ +∞ n(2g(L) + µ -1) ? However this cannot be true. Indeed, there are pairs of mutant knots with dierent genera, and neither of the LG n,1 detects mutation.

The Links-Gould polynomial and fiberedness

In addition to genus information, the Links-Gould polynomial seems to contain signs of whether a knot is bered or not. This is another well known feature of the Alexander invariant. Conjecture 0.4, if it were to be true, would rene the standard Alexander polynomial criterion. This is the object of this section. Denition 3.1. A knot K in S 3 is said to be bered if the two following conditions hold :

1: the complement of the knot is the total space of a locally trivial bundle over the base space S 1 , i.e. there exists a map p : S 3 \ K -→ S 1 which is a locally trivial bundle.

2: there exists V (K) a neighborhood of K and there exists a trivializing homeomor-

phism θ : V (K) -→ S 1 × D 2 such that π • θ(X) = p(X) for any X ∈ V (K) \ K,
where π(x, y) := y |y| . Let us recall well-known properties of the Alexander polynomial of a bered knot. The Alexander polynomial of a bered knot is monic [START_REF] Neuwirth | Knot Groups[END_REF][START_REF] Rapaport | On the commutator subgroup of a knot group[END_REF][START_REF] Stallings | On bering certain 3-manifolds[END_REF]. This means the coecient of the highest degree term of the standard Alexander normalization of the polynomial is 1. For the Conway normalization, it means the leading coecient is ±1. The converse is not true in general. However, the condition is sucient for prime knots with up to 10 crossings and alternating knots [START_REF] Murasugi | On a certain subgroup of the group of an alternating link[END_REF]. Also note that for bered knots, the degree of the Alexander Again, the rst term of this sum has (-2r -4) + 2 + (2q -2) = 2q -2r -4 so we can ignore it as well and we are interested in the following sum :

x(-r)(t -1 0 , t -1 1 )(t -1 0 + t -1 1 -t -1 0 t -1 1 ) + z(-r)(t -1 0 , t -1 1 )
LG(K(p, q, 0))

-x(-r)(t -1 0 , t -1 1 )t -1 0 t -1 1
LG(K(p, q, 1)).

However, span(z(-r)(t -1

0 , t -1 1
)) = -2r-6 and span(x(-r)(t -1

0 , t -1 1 )(t -1 0 +t -1 1 -t -1 0 t -1 1 )) = (-2r -4) + 2 = -2r -2.
So we can reduce our concerns to

x(-r)(t -1 0 , t -1 1 )(t -1 0 + t -1 1 -t -1 0 t -1 1 )
LG(K(p, q, 0))

-x(-r)(t -1 0 , t -1 1 )t -1 0 t -1 1
LG(K(p, q, 1)).

Using the usual skein relations on LG(K(p, q, 0), we get the following value modulo terms with a small span :

LG

(K(p, q, 0) = (t -1 0 t -1 1 ) p/2 -1 t -1 0 t -1 1 -1 -(t -1 0 -1)(t -1 1 -1) x(q)(t 0 + t 1 -1 -t 0 t 1 ) + 2(t -1 0 -1)(t -1 1 -1) t -1 0 t -1 1 -1 p 2 - (t -1 0 t -1 1 ) p/2 -1 t -1 0 t -1 1 -1 x(q) -(t 0 -1)(t 1 -1) .
The two pieces of the sum have span 2q. Similarily, ignoring non extremal span terms :

LG(K(p, q, 1) = (t -1 0 t -1 1 ) p/2 -1 t -1 0 t -1 1 -1

x(q)(t 0 + t 1 -1)(t -1 0 + t -1 1 -2 -t -1 0 t -1 1 ) -(t 0 -1)(t 1 -1)

+ 2(t -1 0 -1)(t -1 1 -1) t -1 0 t -1 1 -1 p 2 - (t -1 0 t -1 1 ) p/2 -1 t -1 0 t -1
1 -1

x(q + 1) -(t 0 -1)(t 1 -1) .

So the quantity we are interested in is

x(-r)(t -1 0 , t -1 )(t -1 0 + t -1 1 -t -1 0 t -1 1 ) (t -1 0 t -1 1 ) p/2 -1 t -1 0 t -1 1 -1 -(t -1 0 -1)(t -1 1 -1) x(q)(t 0 + t 1 ) + 2(t -1 0 -1)(t -1 1 -1)

t -1 0 t -1 1 -1 p 2 - (t -1 0 t -1 1 ) p/2 -1 t -1 0 t -1
1 -1

x(q)(t 0 + t 1 ) + x(-r)(t -1 0 , t -1 1 )(-t -1 0 t -1 1 )

(t -1 0 t -1 1 ) p/2 -1 t -1 0 t -1 1 -1

x(q)(t 0 + t 1 )(t -1 0 + t -1 1 )(t 0 + t 1 )

+ 2(t -1 0 -1)(t -1 1 -1) t -1 0 t -1 1 -1 p 2 - (t -1 0 t -1 1 ) p/2 -1 t -1 0 t -1
1 -1

x(q + 1)(t 0 + t 1 ) = x(-r)(t -1 0 , t 1 -1 )(t 0 + t 1 ) (t -1 0 + t -1 1 ) 2 (t -1 0 t -1 1 ) p/2 -1 t -1 0 t -1 1 -1

x(q)

+ (t -1 0 + t -1 1 ) 2(t -1 0 -1)(t -1 1 -1) t -1 0 t -1 1 -1 p 2 - (t -1 0 t -1 1 ) p/2 -1 t -1 0 t -1
1 -1

x(q) + (t -1 0 t -1 1 ) p/2 -1 t -1 0 t -1 1 -1

x(q)(t 0 + t 1 )(t -1 0 + t -1 1 )(-t -1 0 t -1 1 )

+ (-t -1 0 t -1 1 )

2(t -1 0 -1)(t -1 1 -1) t -1 0 t -1 1 -1 p 2 - (t -1 0 t -1 1 ) p/2 -1 t -1 0 t -1
1 -1

x(q + 1) = x(-r)(t -1 0 , t 1 -1 )(t 0 + t 1 ) x(q) (t -1 0 t -1 1 ) p/2 -1 t -1 0 t -1 1 -1

t -2 0 + t -2 1 -t -1 0 t -1 1 (t 0 t -1 1 + t 1 t -1 0 ) + 2(t -1 0 -1)(t -1 1 -1) t -1 0 t -1 1 -1 p 2 - (t -1 0 t -1 1 ) p/2 -1 t -1 0 t -1
1 -1 (t -1 0 + t -1 1 )x(q) -t -1 0 t -1 1 x(q + 1) .

But t -2 0 + t -2 1 -t -1 0 t -1 1 (t 0 t -1 1 + t 1 t -1 0 ) = 0. So to show the two terms of highest and lowest degree disappear in that polynomial we show that modulo lower degree terms, α = (t -1 0 + t -1 1 )x(q) -t -1 0 t -1 1 x(q + 1) = 0. Let's look at x(q) rst of all :

x(q) = t q 0 -t q 1 + other terms (t 0 + 1)(t 1 + 1)(t 0 -t 1 )

= M + m + other terms, where M is the term of highest degree in x(q), and m the one of smallest degree. That way t q 0 -t q 1 + other terms = (t 0 + 1)(t 1 + 1)(t 0 -t 1 )(M + m + other terms). And identifying the highest and lowest degree terms on each side we nd M = t q-2 0 and m = t q-2 1 .

Finally, modulo lower degree terms, α = (t -1 0 + t -1 1 )(t q-2 0 + t q-2 1 ) -t -1 0 t -1 1 (t q-1 0 + t q-1 1 ) = t -1 1 t q-2 0 -t -1 0 t -1 1 t q-1 0 + t -1 0 t q-2 1 -t -1 0 t -1 1 t q-1 1 = 0.

In conclusion, span(LG(K(p, q, r))) (2q -2r -2) -2 = 2q -2r -4 = 4 q -r -2 2 .

Figure 1 .

 1 Figure 1. The untwisted Whitehead double of the right handed trefoil with a positive clasp.

S 1 : 2 :Figure 2 .

 122 Figure 2. Generators S 1 and S 2 of braid group B 3 .

Remark 2 . 13 .

 213 If m is even, D(b 1 , . . . , b m ) is a knot. It is a link with two components when m is odd. Proposition 2.14. If b i = 0 for any i, then g(D(b 1 , . . . , b m )) = m -µ + 1 2

S 2b 1 2 S -2b 2 1 . . . S -2bm 1 Figure 3 . 2 S -2b 2 1 . . . S 2bm 2 Figure 4 .

 21132124 Figure 3. D(b 1 , . . . , b m ) when m is even.

2

 2 (D(b 1 )) = ã1 (b 1 )LG( ) + ã2 (b 1 )LG( ) + ã3 (b 1 )LG( ) = ã1 (b 1 ) + ã2 (b 1 ). So the span of LG(D(b 1 )) is 2. m = Still using the same skein relation, we can compute LG(D(b 1 , b 2 )).

2n - 1 crossingsFigure 5 .

 15 Figure 5.

Since

  LG(D(b 1 , . . . , b m-1 , 0)) = LG(D(b 1 , . . . , b m-2 )) the rst element in the sum has a span smaller than 0 + 2(m -1) = 2m -2. The second part has span 2 + 2(m -1) = 2m and the third 2 + 2(m -2) = 2m -2. In the end span(LG(D(b 1 , . . . , b m ))) = 2m.

CQFD 2 . 3 . 3 .

 233 Twist knots. Denition 2.16. A twist knot is a Whitehead double of the unknot. We will denote by K n the twist knot shown in Figure5when 2n -1 is positive. If 2n -1 is negative, there are 1 -2n crossings of the opposite sort.

Figure 8 .

 8 Figure 8. K(2, -1, r) when r 3.

Figure 9 .

 9 Figure 9. K(2, -1, r) when r -1.

CQFD 2 . 4 .

 24 A generalization of Conjecture 0.3 to a family of Links-Gould invariants.

  Remark 1.2. Compared to the R-matrix used in

	t 0	.		.	.	.	.		.		.		.	.	.	.	.		.	.	.
	. .	. .		. .	. t .	1/2 0 .	. .		. .		. .	t	. 1/2 0	. .	. .	. .	. .		. .	. .	. .
	.	.		.	.	.	.		.		.		.	.	.	.	1		.	.	.
	. t 1/2 0		.	. t 0 -1 .		.		.		.	.	.	.	.		.	.	.
	.	.		.	.	.	-1		.		.		.	.	.	.	.		.	.	.
	. . . .	. . . .	t	. . 1/2 0 .	. . . .	. . . .	. . . .	t 0 t 1 -1 . . -t 1/2 0 t 1/2 1	. . . .	. . t 0 -1 .	-t 1/2 0 t 1/2 1 . . .	. . . .	. . . .	-t 1/2 0 t . . Y	1/2 1 Y	. 1/2 t 1 . .	. . . .	. . . .
	.	.		.	.	.	.		.		.		.	.	-1	.	.		.	.	.
	. . .	. . .		. . .	. 1 .	. . .	. . -t .	. 1/2 0 t .	1/2 1 Y	. . 1/2 t 1		. . .	. Y .	. . .	. . .	. Y 2 .	. . t 1 -1	t	1/2 1 . .	. . .
	.	.		.	.	.	.		.		.		.	.	. t

  Among these knots we get a more precise lower bound with LG than with ∆ for several of them : 11 N 34 , 11 N 42 , 11 N 67 , 11 N 97 , 12 N 31 , 12 N 51 , 12 N 129 , 12 N 256 , 12 N 257 , 12 N 264 , 12 N 267 , Some of the spans are "half integers" in the sense that they are multiples of 2, as they necessarily are, but not multiples of 4 : 11 N 34 , 11 N 42 , 11 N 67 , 11 N 97 , 12 N 51 , 12 N 256 ,

	11 N 34 , 11 N 42 , 11 N 45 , 11 N 67 , 11 N 73 , 11 N 97 , 11 N 152 , 12 N 28 , 12 N 31 , 12 N 51 , 12 N 56 , 12 N 63 , 129 , 12 N 87 , 12 N 12 N 132 , 12 N 221 , 12 N 231 , 12 N 256 , 12 N 257 , 12 N 264 , 12 N 267 , 12 N 268 , 12 N 313 , 12 N 430 , 12 N 665 , 12 N 808 , 12 N 812 . 4 : 12 N 268 , 12 N 313 , 12 N 430 , 12 N 665 , 12 N 812 . 5 : 12 N 257 , 12 N 264 , 12 N 267 , 12 N 268 , 12 N 313 , 12 N 430 , 12 N 665 , 12 N 812 .
	Remark 2.2.

1/2 1 . . t 1 -1 . . . . . . . . . . . . . .

Acknowledgments. I owe many warm thanks to my Emmanuel Wagner who taught me about Links-Gould invariants and has been of much help in designing and writing this paper. I am most grateful to Hugh Morton for his help during my study of the Whitehead doubles of the trefoil, and for showing me a braid presentation for the untwisted double. I am indebted to David de Wit for the extended work he did implementing the LINKS-GOULD EXPLORER, and to Jon Links for telling me about this remarkable package. Finally I would like to thank David Cimasoni and Anthony Conway for the fruitful and pleasant conversations, as well as for their comments and ideas.

2 : K(p, q, r) has genus 1 is p, q, r are odd and we are not in case 1, 3 : K(±2, ∓1, ±r) has genus |r-2|-1 2 , 4 : K(2l, q, r) has genus |q|+|r| 2 if q, r have the same sign and we are not in any of the previous cases, 5 : K(2l, q, r) has genus |q|+|r|-2 2 if q, r have dierent signs and we are not in cases 1, 2 or 3.

We rewrite that theorem so that dierent cases exclude each other. Doing this makes computations more specic and somewhat easier in each case. Moreover, since K(p, q, r) * = K(-p, -q, -r), we will consider p 0. Also, K(p, q, r) = K(q, r, p) = K(r, p, q). So we can restrict our study to the cases where q, r are odd.

Corollary 2.22. Given the restrictions mentioned, setting p 0 an integer and q, r two odd integers, the genus g of K(p, q, r) is as follows :

1 : g = 0 for K(p, ±1, ∓1) and K(2, -1, 3), that is when K(p, q, r) is the unknot, 2 : g = 1 when p, q, r are odd and K(p, q, r) is not the unknot, 3 :

• p is even and q, r are positive, • p is even and dierent from 2, q = -1 and r is negative, • p is even, q is negative and dierent from -1, r is negative, 5 : g = |q|+|r|-2 2 if :

• p is even and dierent from 2, q = -1 and r 3, • p is even, q 0, r 0 and (p, q, r) = (p, 1, -1), • p is even, q -3 and r 0. Theorem 2.23. For all pretzel knots, span(LG(K(p, q, r))) 4g(K(p, q, r)).

Proof. We compute the span of LG(K(p, q, r)) in each case of Corollary 2.22. 1 K(p, ±1, ∓1) and K(2, -1, 3) are dierent representations of the trivial knot that is part of the small cases we already checked.

2 Using the fact that K(p, q, r) * = K(-p, -q, -r) and K(p, q, r) = K(q, r, p) = K(r, p, q), we have only two cases to consider : when p, q, r have the same sign and when two out of the three have the same sign. For example we can choose the following congurations : p, q, r 0 and p 0, q, r 0. In each case, using skein relation 2 of Corollary 2.12 on the three pairs of strands, we nd a sum of 27 terms, each of which is symmetric of span polynomial is exactly twice the genus of the that is the genus of the corresponding bre surface [START_REF] Rapaport | On the commutator subgroup of a knot group[END_REF]. This last property yields the following. Proposition 3.2. Set K a bered knot. If Conjecture 0.3 is true, then span(LG(K; t 0 , t 1 )) = 4g(K).

Proof. To have such a result we can more generally consider a set of links E such that, for any L ∈ E, deg(∆ L (t)) = 2g(L) + µ -1. This is the case here, and it was also the case in Proposition 1.13 where it is proved completely. CQFD We can express the Links-Gould polynomial with dierent sets of variables : (t 0 , t 1 ) as we did up to now, but also (p, q) where

.

These are the variables used in the LINKS-GOULD EXPLORER as well as in de Wit's papers on the subject. He sometimes uses P = p 2 . In variables (p, q) the LG polynomial of a link L can be written

LG(L; p, q) = a 0 +

where a 0 ∈ Z and P k (q) ∈ Z[q ±1 ]. Note that if P l (q) = 0 and P k (q) = 0 for any k > l, then span(LG(L; p, q)) = 2l.

Denition 3.3. Set K a knot. We say LG(K) is monic when the term in LG(K) of highest and lowest degrees can be written q 2m (p 4l + p -4l ) with l ∈ N and m ∈ Z. In terms of variables (t 0 , t 1 ), this condition is expressed by saying the terms of highest and lowest degrees are monic monomials of the form t α 0 t β 1 with α + β even.

Proposition 3.4. Set K a knot. If LG(K; t 0 , t 1 ) is monic, then ∆ K (t) is monic as well.

Proof. Consequence of LG(K; t, -t -1 ) = ∆ K (t 2 ). CQFD Remark 3.5. Point I in Conjecture 0.4 implies point II. This is a consequence of Proposition 3.4 and of the fact that when the Alexander polynomial is monic for an alternating knot, the knot is bered. Remark 3.6. Given Proposition 3.4, criterion 0.4 would be an improvement of the criterion provided by the Alexander invariant. In addition, we will see in the following that there are examples of knots where ∆ is monic but LG is not. . So ∆ sometimes is monic when LG is not. Verications were done using de Wit's LINKS-GOULD EXPLORER [START_REF] Wit | Links-Gould Explorer[END_REF] and Cha and Livingston's KNOTINFO [START_REF] Cha | KnotInfo: Table of Knot Invariants[END_REF].

Appendix : proof of the harder case Theorem 2.23

Here we prove the remaining case of Theorem 2.23. Consider pretzel knot K(p, q, r) when p is positive and even, q is positive and odd, r is negative and odd, and (q, r) = (1, -1). We want to show that span(LG(K(p, q, r))) 4g = 2q -2r -4.

We rst consider r = -1. Then 4g = 2q -2. In that precise conguration, using skein relation 2 of 2.12 on the p/2 full twists and skein relation 1 of the same corollary on the q half twists, you nd a sum of terms, each of which has a span smaller than 2q -2.

In general, the computation is not that easy. We show knot K(2, 3, -3) in Figure 10 to x the orientation chosen here. Let us introduce some notations :

.

We transform the -r half twists in K(p, q, r) using skein relation 1 of 2.12.

LG(K(p, q, r)

LG(K(p, q, -1)) + z(-r)(t -1 0 , t -1 1 )LG(K(p, q, 0)).

The span of LG(K(p, q, -1)) is 2q-2 with the previous case. Also span(y(-r)(t -1 0 , t -1 1 )) = -2r -4. So the second term of the sum above has span 2q -2r -6 and we need only to consider the rst and third terms in the rest of the proof. Also, using skein relation 1 of Proposition 2.7,

x(-r)(t -1 0 , t -1 1 )LG(K(p, q, -2)) = x(-r)(t -1 0 , t -1 1 )(t -1 0 + t -1 1 -1)LG(K(p, q, -1)) + x(-r)(t -1 0 , t -1 1 )(t -1 0 + t -1 1 -t -1 0 t -1 1 )LG(K(p, q, 0)) -x(-r)(t -1 0 , t -1 1 )t -1 0 t -1 1 LG(K(p, q, 1)).