Adrien Barbaresi

Toponyms as Entry Points into a Digital Edition: Mapping Die Fackel (1899-1936)

Introduction

The significance of place names exceeds the usually admitted frame of deictic and indexical functions, as they enfold more than a mere reference in space. In the western tradition, a current of reflexion which seems to date back to the 1960s has provided the theoretical foundations of the "spatial turn", whose epitome is the concept of space as emergent rather than existing a priori, and composed of relations rather than structures [START_REF] Warf | Introduction: the reinsertion of space into the social sciences and humanities[END_REF]. The emergence of current named "GeoHumanities" [START_REF] Dear | GeoHumanities: Art, History, Text at the Edge of Place[END_REF] or "Spatial Humanities" (Bodenhammer et al., 2010), has prompted for a transfer of research objects between disciplines as well as an enforcement of the spatial turn in practice through specific methods of analysis. The common denominator consists in opening up new spaces and experimenting in a transdisciplinary perspective [START_REF] Domínguez | Literary Geography and Comparative Literature[END_REF] in a field which has been evolving at an exponential pace since the last decade [START_REF] Caquard | Narrative Cartography: From Mapping Stories to the Narrative of Maps and Mapping[END_REF].

In this paper, I introduce a visualization of collocations of toponyms in the satirical literary magazine Die Fackel ("The Torch"), originally published and almost entirely written by the satirist and language critic Karl Kraus in Vienna from 1899 to 1936. This work carries heterogeneity at its core and contains a considerable variety of toponyms [START_REF] Biber | In Wien, in Prag und infolgedessen in Berlin -Ortskonstellationen in der 'Fackel[END_REF] which are highly significant because of the multinational nature of the Austro-Hungarian empire and the later formation of a territorially diminished state.

In order to provide an additional, synthetic access to a digital edition of the work which is already available online (AAC-Fackel corpus1 ), I set out on a distant reading experiment leading to maps meant to uncover patterns and specificities which are not easily retraceable during close reading. I focus on the concept of visualization, that is on the processes and not on the products [START_REF] Crampton | Maps as social constructions: power, communication and visualization[END_REF], and present them together with a critical apparatus, by giving a theoretical perspective on what is being shown and seen. In fact, digital methods in humanities ought to be criticized [START_REF] Wulfman | The Plot of the Plot: Graphs and Visualizations[END_REF] and the cartographic enterprise bears both a thrill and a risk: "adding more to the world through abstraction", and "adding to the riskiness of cartographic politics by proliferating yet more renders of the world" [START_REF] Gerlach | Lines, contours and legends. Coordinates for vernacular mapping[END_REF].

Extraction of toponyms

The particular task of finding place names in texts is commonly named place names extraction, toponym resolution, or geocoding. A first stage involves the identification of potential geographic references, while a second stage resides in a disambiguation process [START_REF] Leetaru | Fulltext geocoding versus spatial metadata for large text archives: Towards a geographically enriched Wikipedia[END_REF]. Toponym resolution often relies on named-entity recognition and artificial intelligence [START_REF] Leidner | Detecting Geographical References in the Form of Place Names and Associated Spatial Natural Language[END_REF]. However, knowledge-based methods using fine-grained data -for example from Wikipedia -have already been used with encouraging results (Hu et al., 2014).

The present endeavor grounds on a specially curated gazetteer: during the 20th century there have been significant political changes in Central Europe that have severely affected toponyms, so that geographical databases lack coverage and detail. Consequently, the database developed at the Austrian Academy of Sciences (Academy Corpora) in cooperation with the Berlin-Brandenburg Academy of Sciences (Language Center) focuses on Europe and follows from a combination of approaches: gazetteers are curated in a semi-supervised way to account for historical differences, and current geographical information is used as a fallback. 2The tokenized files of works to be analyzed are filtered and matched with the database by finitestate automatons [START_REF] Barbaresi | Extraction and Visualization of Toponyms in Diachronic Text Corpora[END_REF]: toponyms (single or multi-word expressions) are extracted using a sliding window. A cascade of filters is used: current and historical states; regions, important subparts of states, and regional landscapes; populated places; and geographical features. Disambiguation being a critical component [START_REF] Leetaru | Fulltext geocoding versus spatial metadata for large text archives: Towards a geographically enriched Wikipedia[END_REF], an algorithm similar to Pouliquen et al. (2006), who demonstrated that an acceptable precision can be reached that way, guesses the most probable entry based on distance to Vienna (Sinnott, 1984), contextual information (closest-country, last names resolved), and importance (place type, population count). The results are projected on a map of Europe using TileMill.3

From collocations to lines of thought

In a further analysis, I visualize co-occurrences of extracted toponyms, which can be considered to be a subset of GeoCollocations [START_REF] Bubenhofer | Geokollokationen -Diskurse zu Orten: Visuelle Korpusanalyse[END_REF], in order to draw sequences, airborne lines following their order of appearance. The word "network" is to be used with circumspection as [START_REF] Latour | On recalling ANT[END_REF] suggests. Although it is ubiquitous in the terminology of the spatial turn, the now predominant interpretation in the sense of the World Wide Web suggests an immediacy which is contrary to the acceptions it had before, so that the concept of "meshwork" is more appropriate [START_REF] Ingold | Lines: A Brief History[END_REF]. I thus interpret Figure 1 as a general meshwork which makes it possible to visualize paths depicting chains of thought (Gedankengänge) as well as their intensity (well-trodden or seldom). If they may reveal spatial patterns that would otherwise remain hidden in texts (Bodenhammer et al., 2010), these linkages are also "mappings and tracing imposed on the data" [START_REF] Wulfman | The Plot of the Plot: Graphs and Visualizations[END_REF] which are not meant to be interpreted without further filtering. That is why I refine the map by selecting a subset of the co-occurrences -the maximal distance between two extracted place names is fixed to twenty tokens -and by color-coding qualitative features -the typology of place names and the axis of time. The most frequent place names are printed out. Surfaces (regions for instance) cannot be represented as such because of historical evolutions and because of the difficulties of linking surfaces without tampering with map readability. Coastlines are depicted in gray to give a sense of orientation, no boundaries are drawn, as they are of a changing nature and may superimpose an artificial reading of the map. 4 The notion of rhizome has been used in corpus linguistics by [START_REF] Scharloth | Das Wuchern der Rhizome. Linguistische Diskursanalyse und Data-driven Turn[END_REF] to qualify discourses captured by collocation graphs, it has originally been coined by [START_REF] Deleuze | Mille Plateaux[END_REF][START_REF] Deleuze | Mille Plateaux[END_REF]). This concept is particularly adequate for Kraus, as the Austrian satirist has always 4 "Territorial maps of ancient states are an idealized projection of state authority rather than a depiction of the way in which ancient political domains were actually governed" (Smith 2005) been concerned by the multiple aspects of discourse and reality. In addition, his work in Die Fackel evades distant reading processes because of the number of citations used and its ever present and extensive usage of parody. It would be vain to design an authoritative cartography of Die Fackel: following from the principles of heterogeneity and "asignifying rupture" (ibid.), the lines are frequently interrupted. Phenomena in the low-frequency range are filtered out by the human eye, but clusters and interpretation cues may emerge which provide a different access to the work. In this regard, Figure 2 depicts a rhizome connecting heterogeneous information, just as we are all "traversed by lines, geodesics, tropics, and zones marching to different beats and differing in nature" (ibid.).

A rhizome as entry to Die Fackel

Conclusion

I have presented a distant reading experiment which consists in connecting toponyms extracted and projected on maps. The latter are meant to be released as an additional feature to the existing digital edition. 5 The first example displays unfiltered lines of thought, whereas the second one grounds on a refined analysis and lets the practical image of a rhizome emerge. While Die Fackel criticizes mechanical, instrumental language [START_REF] Hirt | Improving Wikipedia-Based Place Name Disambiguation in Short Texts Using Structured Data from Dbpedia[END_REF], the "well-informed" linguistic instruments can help materializing dots or sequences, but not without "human" intervention. The filtering steps on the projection echo the hermeneutic circle of the philological tradition; they also make computed information visible and apprehensible which could remain "blind" otherwise [START_REF] Barbaresi | La Raison aveugle ? L'époque cybernétique et ses dispositifs[END_REF]. This is not an authoritative cartography of Die Fackel but rather an indirect depiction of the viewpoint of Kraus and his contemporaries. Drawing on Kraus' vitriolic recording of political life, toponyms in Die Fackel tell a story about the ongoing reconfiguration of Europe. As the map conveys the uncanny sensation that the continent is a field on which points east and west are projected, the lines of force entangle European countries and capitals. Their spatial patterns document an inclination for major cultural centers, whereas the chronological dimension captures a major shift towards the end of publication: the force field intensifies as its range narrows, showing both the interplay of major European powers of the time and the emergence of transatlantic (westwards) and transeuropean (eastwards) relationships. This evolution can be read as an intensification of tensions and a prefiguration of other schemes, this time of military nature.

Figure 1 :

 1 Figure 1: Unfiltered map of toponymic co-occurrences

Figure 2 :

 2 Figure 2: Refined analysis (size proportional to corpus frequency; yellow: sovereign territories; orange: regions; green: populated places; blue: geographical features; time axis represented by a gradient from light green to dark blue)

http://aac.ac.at/fackel Free online access to 37 volumes, 415 issues, 922 numbers; more than

22.500 pages and 6 million wordforms.

Wikidata API (https://www.wikidata.org) and the Geonames database (http://www.geonames.org) are used to build the database semi-automatically.

https://www.mapbox.com/tilemill/


Motaz Saad
email: motaz.saad@loria.fr

David Langlois
email: david.langlios@loria.fr

Kamel Smaïli
email: kamel.samili@loria.fr

Cross-Lingual Semantic Similarity Measure for Comparable Articles

Keywords: cross-lingual latent semantic indexing, corpus comparability, cross-lingual information retrieval

published or not. The documents may come

Introduction

Comparing cross-lingual articles is a challenging issue in several topics in natural language processing and especially in machine translation and cross-lingual information retrieval. The comparison can be done in terms of topics, opinions, or emotions. In this paper, we focus on how to retrieve comparable articles, for that, we need a specific measure. A comparable corpus is a collection of articles, in multiple languages, which are not necessarily translations of each other, but they are related to the same topic. In some sense, a parallel corpus can be considered as comparable in which each sentence in the source corpus is aligned with its translation in a target corpus.

There are many proposed methods to compare or retrieve cross-lingual articles. These methods are based on bilingual dictionaries [START_REF] Li | Improving corpus comparability for bilingual lexicon extraction from comparable corpora[END_REF][START_REF] Otero | Measuring comparability of multilingual corpora extracted from wikipedia[END_REF][START_REF] Saad | Extracting comparable articles from wikipedia and measuring their comparabilities[END_REF], or based on cross-lingual Information retrieval (CL-IR) [START_REF] Fujii | Applying machine translation to two-stage cross-language information retrieval[END_REF][START_REF] Aljlayl | On Arabic-English Cross-Language Information Retrieval: Machine Translation Approach[END_REF][START_REF] Ture | Searching to Translate and Translating to Search: When Information Retrieval Meets Machine Translation[END_REF], or based on cross-lingual Latent Semantic Indexing (CL-LSI) system [START_REF] Berry | Using latent semantic indexing for multilanguage information retrieval[END_REF][START_REF] Littman | Automatic cross-language information retrieval using latent semantic indexing[END_REF][START_REF] Dumais | Lsa and information retrieval: Getting back to basics[END_REF][START_REF] Muhic | Cross-lingual document similarity[END_REF].

In the dictionary based method [START_REF] Li | Improving corpus comparability for bilingual lexicon extraction from comparable corpora[END_REF][START_REF] Otero | Measuring comparability of multilingual corpora extracted from wikipedia[END_REF][START_REF] Saad | Extracting comparable articles from wikipedia and measuring their comparabilities[END_REF], two cross-lingual documents d a and d e are comparable if a maximum of words in d a are translations of words in d e , so a bilingual dictionary can be used to look-up the translation of words in both documents. The drawbacks of this approach are the dependency on bilingual dictionaries which are not always available, and the necessity to use morphological analyzers for languages that can be inflected. Moreover, word-toword translations based on dictionaries can cause many errors. [START_REF] Saad | Extracting comparable articles from wikipedia and measuring their comparabilities[END_REF] proposed binary and cosine measures based on multi-WordNet [START_REF] Bond | A survey of wordnets and their licenses[END_REF] dictionary to compare Wikipedia and news articles. Both binary and cosine measures proposed by [START_REF] Saad | Extracting comparable articles from wikipedia and measuring their comparabilities[END_REF] require the source-target texts to be represented as vectors of aligned words. Word weight for the binary measure is either 1 or 0 (presence or absence of the word), while it is the term frequency for the cosine measure. The similarity of cross-lingual documents is computed as follows: the binary measure counts the words in d a which are translation of words in d e , then it normalizes by the vector size, while the cosine measure computes the cosine between source and target vectors which represent the frequency of the aligned words of d a and d e .

In the Cross-Lingual Information Retrieval (CL-IR) method, one can use Machine Translation (MT) systems in order to achieve source and target documents into the same language; then classical IR tools can be used to identify comparable articles [START_REF] Fujii | Applying machine translation to two-stage cross-language information retrieval[END_REF][START_REF] Aljlayl | On Arabic-English Cross-Language Information Retrieval: Machine Translation Approach[END_REF][START_REF] Ture | Searching to Translate and Translating to Search: When Information Retrieval Meets Machine Translation[END_REF]. Query documents are usually translated into the language of indexed documents, this is because the computational cost of translating queries is less than the cost of translating the whole indexed documents. The drawback of this approach is the dependency on MT systems, so the performance of the MT affects the performance of the IR system. Moreover, the MT system needs to be developed first if it is not available for the desired languages.

In Cross-Lingual Latent Semantic Indexing (CL-LSI) method, documents are described as numerical vectors which are mapped into a new space, then one can compute the cosine between vectors to measure the similarity between them. The LSI method has yet been used in the scope of CL-IR by [START_REF] Berry | Using latent semantic indexing for multilanguage information retrieval[END_REF][START_REF] Littman | Automatic cross-language information retrieval using latent semantic indexing[END_REF][START_REF] Muhic | Cross-lingual document similarity[END_REF]. In their approach, the source document and its translation (the target) are concatenated into one document, then the LSI makes links between source and target words or documents. [START_REF] Berry | Using latent semantic indexing for multilanguage information retrieval[END_REF] Focused their work on Greek-English document retrieval, while [START_REF] Littman | Automatic cross-language information retrieval using latent semantic indexing[END_REF] focused on French-English documents, and [START_REF] Muhic | Cross-lingual document similarity[END_REF] computed the similarity of Wikipedia articles in several European languages.

In this work, we focus on CL-IR for English-Arabic document retrieval. In order to avoid using bilingual dictionaries or morphological analyzers or MT systems, we use CL-LSI to compare and retrieve English-Arabic documents. Another advantage for CL-LSI is that it overcomes the problem of vocabulary mismatch between queries and documents. So, we use the same approach as [START_REF] Littman | Automatic cross-language information retrieval using latent semantic indexing[END_REF] but we apply it on Arabic-English articles, moreover, [START_REF] Littman | Automatic cross-language information retrieval using latent semantic indexing[END_REF] used parallel corpus in their work, but we use both parallel and comparable corpus to train the CL-LSI.

In this paper, we use LSI in two ways to retrieve Arabic-English comparable documents. The first one is monolingual: the English article is translated and then mapped into the LSI Arabic space; the second one is cross-lingual: Arabic and English articles are mapped into Arabic-English CL-LSI space. We also compare these methods to the dictionary based method proposed by [START_REF] Saad | Extracting comparable articles from wikipedia and measuring their comparabilities[END_REF] which is described above.

Besides using the CL-LSI to retrieve comparable articles, we also use it to measure the "comparability of a corpus" which is to inspect if a target corpus is a translation of a source one, and how much they are different from each others. This permits to learn how much two comparable corpora are similar to each others. This can be useful for many applications such as cross-lingual lexicon extraction, information extraction, and sentence alignment.

The rest of the paper is organized as follows: corpora and the method are described in Sections 2, 3, and 4. Results are presented and in Section 5. Finally, the conclusion is stated.

Corpora

In this section we describe the material we use for our different experiments. It is constituted from documents collected from newspapers, United Nations resolutions, talks, movie subtitles and other domains. These corpora are either parallel or comparable. In the following sections, we describe these corpora.

Parallel Corpora

Table 1 presents the parallel corpora, where |S| is the number of sentences, |W | is the number of words, and |V | is the vocabulary size. The table also shows the domain of each corpus.

The parallel corpora that we use are: AFP1 , ANN2 , ASB3 [START_REF] Ma | Arabic newswire english translation collection[END_REF], Medar4 , NIST [START_REF] Nist | NIST 2008/2009 open machine translation (OpenMT) evaluation. Linguistic Data Consortium[END_REF], UN [START_REF] Rafalovitch | United nations general assembly resolutions: A sixlanguage parallel corpus[END_REF], TED 5 [4], OST6 [START_REF] Tiedemann | Parallel data, tools and interfaces in opus[END_REF], and Tatoeba7 [START_REF] Tiedemann | Parallel data, tools and interfaces in opus[END_REF].

Note that OST is a collection of movie subtitles translated and uploaded by users. So, the quality of the translations may vary from a user to another.

It can be noted from Table 1, in all parallel corpora, English texts have more words than Arabic; in contrast, Arabic texts have vocabularies larger than English. The reason is that certain Arabic terms can be agglutinated [START_REF] Meftouh | Modeling Arabic Language using statistical methods[END_REF], while English terms are isolated. For instance, the Arabic item wasano t . eyhm which corresponds to "and we will give them" in English, is an example of one Arabic term that corresponds to five English words. On the other hand, Arabic has a larger vocabulary because, it is morphologically rich [START_REF] Habash | Introduction to Arabic natural language processing[END_REF][START_REF] Saad | The Impact of Text Preprocessing and Term Weighting on Arabic Text Classification[END_REF]. For example, the English word "travellers" may correspond to three forms in Arabic: mosāferwn in masculine nominative form, mosāferyn in masculine accusative/genitive form, or mosāferāt in feminine form. and EuroNews is collected from EuroNews website.9 WIKI articles are edited online by Wikipedia community. There is a hyperlink between articles that are related to the same topic, but each article may be written independently. Therefore, Wikipedia articles are not necessarily translations of each other.

LSI-based Methods

The LSI method [START_REF] Deerwester | Indexing by latent semantic analysis[END_REF] decomposes the term-document matrix X into:

X = U SV T .
The decomposition is done by the Singular Value Decomposition (SVD). The matrices U and V T are the left and right singular vectors respectively, while S is a diagonal matrix of singular values. Each column vector in the matrix U maps terms in the corpus into a single concept, where semantically related terms are grouped with similar values in U . The decomposition U SV T has a rank R, where R is the reduced number of concept dimension in LSI.

For monolingual LSI approach, X is represented as in [START_REF] Aljlayl | On Arabic-English Cross-Language Information Retrieval: Machine Translation Approach[END_REF]. It is a m×n matrix that represents a given monolingual corpus which consists of n documents, and m terms. The entries w ij are the tf idf weights.

X =      d 1 d 2 . . . d
     (1)
X =               d u 1 d u 2 . . .
              (2)
In cross-lingual LSI approach, X is represented as in [START_REF] Berry | Using latent semantic indexing for multilanguage information retrieval[END_REF]. Each d u i is the concatenation of the Arabic document d a i and its corresponding English document d e i . Consequently, X represents a bilingual corpus consisting of n cross-lingual documents, l Arabic terms, and m English terms. So, X is a (l + m) × n matrix. X as represented in (2) can be used to represent parallel or comparable corpora. For parallel corpus, each d u i represents a pair of parallel sentences, while for comparable corpus, it represents a pair of comparable documents. Describing corpus as formulated in [START_REF] Berry | Using latent semantic indexing for multilanguage information retrieval[END_REF], enables LSI to learn the relationship between terms which are semantically related in the same language or between two languages.

So, we use this method to achieve our objective which is to retrieve comparable articles. We describe in the next section how to do that.

Experiment Procedure

As outlined in the introduction, for a source document (English), we want to retrieve target comparable documents (Arabic). So, the source document is compared with all target documents, then the most similar target documents are retrieved. This is done by describing the source and target documents as bag-ofwords, then mapping them into vectors in LSI space, and then by computing the angle between these vectors. If the cosine value between the two vectors is high, then we consider these two documents as comparable. All English and Arabic texts are preprocessed by just removing punctuation marks.

In the next sections, we describe how LSI matrices are built, and how they are used to retrieve comparable articles. Then we compare the results of these two methods.

Building LSI Matrices

Steps below describe how LSI matrices are built:

1. Split English and Arabic corpora presented in Section 2 into training (90%) and testing (10%). 2. Use Arabic training corpus to create X as in [START_REF] Aljlayl | On Arabic-English Cross-Language Information Retrieval: Machine Translation Approach[END_REF]. Then apply LSI to obtain U SV T , this will achieve the monolingual LSI matrix (LSI-AR) as described in left side of the Figure 1. 3. Use English-Arabic training corpus to create X as in [START_REF] Berry | Using latent semantic indexing for multilanguage information retrieval[END_REF]. Then apply LSI to obtain U SV T , this will achieve the cross-lingual LSI matrix (LSI-U) as described in right side of the Figure 1. The optimal rank of U SV T in steps 2 and 3 above is chosen experimentally. According to [START_REF] Landauer | An introduction to latent semantic analysis[END_REF], the optimal number of dimensions to perform the SVD is in the range [100 . . . 500]. We conducted several experiments in order to determine the best rank, and we found that 300 is the dimension which optimizes the similarity for the parallel corpus. So, we use the dimension 300 in all our experiments.

Retrieving Comparable Articles

The test corpus is composed of n couples of English e i and Arabic a j documents (aligned at sentence level in parallel corpus and at the document level in comparable corpus). The goal is then to retrieve the a i among all the a j given e i . The following steps describe the two used methods.

LSI-AR:

1. For each a j , get a j : a j = a t j U S -1 . 2. Translate each English document e i into Arabic using Google MT service 10and get a ei . 3. For each a ei , get a ei : a ei = a t ei U S -1 . 4. For each a ei and a j compute cosine(a ei , a j ).

LSI-U:

1. For each a j , get a j : a j = a t j U S -1 . 2. For each e i , get e i : e i = e t i U S -1 . 3. For each e i and a j compute cosine(e i , a j ).

Where e i , a ei , and a j are vectors of the same nature since they have a language independent representation. Now we can use the cosine values to get the most similar Arabic document to a given English one. For each e i , we sort a j in descending order according to the cosine values. e i and a j are truly comparable if i = j. In other words, for each source document, we have only one relevant document. So, in the sorted list of a j , the condition (i = j) is checked in the top-1 (recall at 1 or R@1), top-5 (recall at 5 or R@5), and top-10 (recall at 10 or R@10) lists. The performance measure is defined as the percentage of a i which are truly retrieved in R@1, R@5, R@10 lists, among all e i .

Results and Discussion

Retrieving Parallel Articles

Results of LSI-AR and LSI-U approaches are presented in Table 3. Results are presented for a random sample of 100 source and target test articles because of the computational cost of doing the experiment on all the test corpus. As shown in Table 3, it is difficult to get a general conclusion about the performance of LSI since it depends on the nature of the corpus and on the desired recall (R@1, R@5 or R@10). For example, for AFP, ASB, TED, UN, and Medar, LSI-U is slightly better than LSI-AR. In contrast, for ANN, NIST, OST and Tatoeba, LSI-AR is better than LSI-U. The performance of LSU-U is equal to, or better than LSI-AR in 6 over 9 of corpora for R@1. The average value for (R@1) in LSI-AR and LSI-U methods are 0.71 and 0.72 respectively. Moreover, we checked the significance of these differences (McNemar's test), and we found that they are not significantly different. Therefore, both approaches obtain mostly the same performance. In addition, we recall that the LSI-U does not require a MT system. Therefore, we can affirm that the LSI-U is competitive compared to LSI-AR.

The performance of LSI-AR and LSI-U approaches on OST corpus is poor as expected because of the nature of this corpus, which is composed of subtitles that are translated by many users as mentioned in Section 2. To investigate the effect of the performance of the MT system on the performance of the LSI-AR, we run an experiment to simulate a perfect system. This is done by retrieving an Arabic document by providing the same document as a query. This experiment is done on all corpora, and the results in terms of R@1 are 1.0 for all corpora. These results reveal the lack of robustness of LSI-AR according to the MT system's performance.

We compare our method with the dictionary based method that was proposed by [START_REF] Saad | Extracting comparable articles from wikipedia and measuring their comparabilities[END_REF] on the union of AFP and ANN corpora. Results are presented in Table 4 where the dictionary based method is denoted as DICT.

As can be noted in the table, both LSI methods achieve better results than DICT, except for R@10 which is slightly better for DICT. It can be concluded that this method is better than DICT since it does not need any dictionary nor morphological analysis and it is language independent. Table 4. Recall results the union of AFP and ANN corpora Method R@1 R@5 R@10 DICT 0.49 0.81 1.0 LSI-AR 0.87 0.95 0.96 LSI-U 0.86 0.96 0.98

Retrieving Comparable Articles

For comparable corpora, the same experimental protocol is applied. Table 5 shows the performance of recall of the LSI-U method on EuroNews and WIKI comparable corpora. As shown in the table, the performance of the LSI-U on EuroNews corpus is better than WIKI corpus. This could be due to the fact that EuroNews articles are mostly translations of each other, while Wikipedia articles are not necessarily translations of each other as mentioned in Section 2.

From Tables 5 and3, it can be noted that LSI-U can retrieve the target information in respectively document level and sentence level with almost the same performance since for parallel corpora AFP, ANN, and ASB, R@1 achieved 0.97, 0.83, and 0.84 respectively, and for the comparable corpus EuroNews, R@1 got 0.84.

Comparing Corpora

We take advantage of the used method, in order to study the comparability of some supposed comparable corpora such as WIKI, EuroNews. We do that by computing the average cosine avg(cos) for all pair articles of the test parts of these corpora. So, for each corpus, the LSI-U matrix is built from the training part, and used to compute the avg(cos) for the test part. This experiment is done on BEST, EuroNews, and WIKI corpora. BEST is the union of AFP, ASB, and UN parallel corpora. These corpora are chosen because they have the best recall performance as shown in Table 3. Statistics on comparability are presented in Table 6.

The average similarity proposes to corroborate the fact that for the parallel corpus, we get better recall results than the others. In other words, the score for BEST which is a parallel corpus aligned at sentence level is better than the one for WIKI which is considered as a real comparable corpus, and for EuroNews (near parallel), which is composed of translated articles, the results are better than for WIKI, but lower than for BEST.

Conclusion

We used in this paper a method which permits to measure the comparability between corpora. This method is based on LSI which we used in two ways: monolingual (LSI-AR) and cross-lingual (LSI-U). The first method needs to use a machine translation system in order to compare two vectors of the same kind of data, whereas the second method merges the training data of both languages and in the test step the comparison is then done on two vectors of the same type since they contain the representation of two cross-lingual documents.

We applied this method on English-Arabic documents. The method allows us to identify comparable articles extracted from a variety of corpora. The measure we proposed has shown its feasibility since it permits to distinguish the parallel corpora from the strongly comparable corpora such as Euronews, and also from the weakly comparable corpora such as WIKI. The feasibility of the method has been illustrated in this paper since it has been tested on 9 different corpora. Some of them are largely used by the community and others are less popular but more difficult such as OST. The best results have been achieved for AFP corpus and the worst for OST. In a future work, we will use this method in order to retrieve comparable articles from the social media to collect and build parallel corpora for languages which are under-resourced such as vernacular ones. The method developed in this paper will be deepened and adapted in order to compare the cross-lingual corpora in terms of opinions and emotions.

Fig. 1 .

 1 Fig. 1. LSI models

Table 1 .

 1 Parallel Corpora

	Corpus	|S|	|W | English Arabic	|V | English Arabic
	Newspapers
	AFP	4K	140K 114K	17K 25K
	ANN	10K	387K 288K	39K 63K
	ASB	4K	187K 139K	21K 34K
	Medar	13K	398K 382K	43K 71K
	NIST	2K	85K 64K	15K 22K
	United Nations Resolutions
	UN	61K	2.8M 2.4M	42K 77K
	Talks
	TED	88K	1.9M 1.6M	88K 182K
	Movie Subtitles
	OST	2M	31M 22.4M	504K 1.3M
	Other
	Tatoeba	1K	17K 13K	4K	6K
	Total	2.3M	37M 27.5M	775K 1.8M
	2.2 Comparable Corpora

Table 2

 2

shows WIKI and EuroNews comparable corpora, where |D| is the number of articles, |W | is the number of words, and |V | is the vocabulary size. Each pair of comparable articles is related to the same topic. WIKI and EuroNews were collected and aligned at article level by

[START_REF] Saad | Extracting comparable articles from wikipedia and measuring their comparabilities[END_REF]

. WIKI is collected from Wikipedia 8

Table 2 .

 2 Comparable Corpora

		WIKI	EuroNews
		English Arabic	English Arabic
	|D|	40K 40K	34K 34K
	|W |	91.3M 22M	6.8M 5.5M
	|V |	2.8M 1.5M	232K 373K

Table 3 .

 3 LSI results for parallel corpora

	Corpus Method R@1 R@5 R@10
	Newspapers
	AFP	LSI-AR 0.94 0.96 0.99 LSI-U 0.97 0.99 0.99
	ANN	LSI-AR 0.80 0.91 0.94 LSI-U 0.82 0.92 0.94
	ASB	LSI-AR 0.79 0.90 0.92 LSI-U 0.85 0.92 0.97
	Medar	LSI-AR 0.56 0.76 0.81 LSI-U 0.61 0.78 0.85
	NIST	LSI-AR 0.78 0.87 0.92 LSI-U 0.71 0.82 0.84
	United Nations Resolutions
	UN	LSI-AR 0.97 1.00 1.00 LSI-U 0.98 0.99 1.00
	Talks
	TED	LSI-AR 0.52 0.73 0.82 LSI-U 0.60 0.83 0.92
	Movie Subtitles
	OST	LSI-AR 0.39 0.61 0.72 LSI-U 0.33 0.76 0.85
	Other
	Tatoeba	LSI-AR 0.70 0.85 0.94 LSI-U 0.61 0.79 0.86

Table 5 .

 5 Testing LSI-U on comparable corpora

	Corpus	R@1 R@5 R@10
	WIKI	0.42 0.84 0.94
	EuroNews 0.84 0.99 1.0

Table 6 .

 6 Statistics on comparability

	Corpus BEST EuroNews WIKI
	avg(cos) 0.53	0.46	0.23

www.afp.com

www.annahar.com

www.assabah.com.tn

www.medar.info

www.ted.com

www.opensubtitles.org

www.tatoeba.org

www.wikipedia.org

www.euronews.com

translate.google.com


Pierre Dehornoy

Ana Rechtman

The trunkenness of a volume-preserving vector field

published or not. The documents may come   L'archive ouverte pluridisciplinaire

The problem we address here is the construction of new invariants of volume-preserving vector fields on S 3 or on compact domains of R 3 up to volume-preserving diffeomorphisms. This problem is motivated by at least two physical situations. First if v is the velocity field of a time-dependant ideal fluid satisfying the Euler equations (ideal hydrodynamics) then its vorticity field curl v is transported by the flow of v [Hel1858]. Second if B is the magnetic field in an incompressible plasma (ideal magnetodynamics), then B turns out to be transported by the velocity field as long as the latter does not develop singularities [START_REF] Lodewijk | A Theorem on Force-free magnetic Fields[END_REF]. In these contexts, invariants of curl v or B up to volume-preserving diffeomorphisms yield time-independent invariants of the system.

Not so many such invariants exist. The first one was discovered by William Thomson [Tho1867]: if the considered field has a periodic orbit or a periodic tube, then its knot type is an invariant (this remark led to the development of knot theory by Peter G. Tait [Tai1877]). However it may not be easy to find periodic orbits, and even then such an invariant only takes a small part of the field into account.

The main known invariant is called helicity. It is defined by the formula Hel(v) = v • u, where u = curl -1 (v) is an arbitrary vector-potential of v. It was discovered by Woltjer, Moreau, and Moffatt [START_REF] Lodewijk | A Theorem on Force-free magnetic Fields[END_REF][START_REF] Moreau | Constantes d'un îlot tourbillonnaire en fluide parfait barotrope[END_REF][START_REF] Keith | The degree of knottedness of tangle vortex lines[END_REF]. Helicity is easy to compute or to approximate since it is enough to exhibit a vector-potential of the considered vector field, to take the scalar product and to integrate. The connection with knot theory was sketched by Moffatt [START_REF] Keith | The degree of knottedness of tangle vortex lines[END_REF] and deepened by Arnold [START_REF] Vladimir | The asymptotic Hopf invariant and its applications[END_REF] as follows. Denote by k X (p, t) a loop starting at the point p, tangent to the vector field X for a time t and closed by an arbitrary segment of bounded length. Denote by Lk the linking number of loops. Arnold showed that for almost every p 1 , p 2 , the limit lim t1,t2→∞

1 t 1 t 2 Lk(k X (p 1 , t 1 ), k X (p 2 , t 2 ))
exists (see also [START_REF] Vogel | On the asymptotic linking number[END_REF] for a corrected statement). Moreover if X is ergodic the limit coincides almost everywhere with Hel(X) (for a non-ergodic vector field, one has to average the previous limit).

The idea of considering knot invariants of long pieces of orbits of the vector field was pursued by Gambaudo and Ghys [START_REF] Jean | Signature asymptotique d'un champ de vecteurs en dimension 3[END_REF] who considered ω-signatures of knots, Baader [START_REF] Baader | Asymptotic concordance invariants for ergodic vector fields[END_REF] who considered linear saddle invariants, and Baader and Marché [START_REF] Sebastian | Asymptotic Vassiliev invariants for vector fields[END_REF] who considered Vassiliev's finite type invariants. In every case, it is shown that lim t→∞ 1 t n V (k X (p, t)) exists, where V is the considered invariant and n a suitable exponent called the order of the asymptotic invariant. However all these constructions have the drawback that they do not yield any new invariant for ergodic vector fields, as in this case the obtained limits are all functions of the helicity.

Recently, it was proved by Kudryavtseva [START_REF] Elena | Helicity is the only invariant of incompressible flows whose derivative is continuous in C 1 -topology[END_REF] for vector fields obtained by suspending an area-preserving diffeomorphism of a surface and for non-vanishing vector fields, and then by Enciso, Peralta-Salas and Torres de Lizaur [START_REF] Alberto | Helicity is the only integral invariant of volume-preserving transformations[END_REF] for arbitrary volume-preserving vector fields, that every invariant that is regular integral (in the sense that its Fréchet derivative is the integral of a continuous kernel) is a function of helicity, see the cited articles for precise statements. These results give a satisfactory explanation of why most constructions yield invariants that are functions of helicity for ergodic vector fields. However they do not rule out the existence of other invariants, but imply that such invariants cannot be too regular.

An example of such another invariant is the asymptotic crossing number considered by Freedman and He [START_REF] Michael | Divergence-Free Fields: Energy and Asymptotic Crossing Number[END_REF]. The advantage is that it is not proportional to helicity, but the disadvantage is that it is hard to compute, even on simple examples.

In this article we consider a less known knot invariant called the trunk (see Definition 1.1 below). It was defined by Ozawa [START_REF] Makoto | Waist and trunk of knots[END_REF], building on the concept of thin position that was introduced by Gabai [START_REF] David | Foliations and the topology of 3-manifolds III[END_REF] for solving the R-conjecture. Less famous that the invariants previously studied in the context of vector fields, the trunk has the advantage that its definition relies on surfaces transverse to the considered knot, so that it is easy to transcript in the context of vector fields. The invariant depends on an invariant measure for the flow of the vector field that may or may not be a volume, and is invariant under diffeomorphisms that preserve this measure. Given a µpreserving vector field X and a surface S, the geometric flux through S is the infinitesimal volume that crosses S in both directions (see Definition 2.1), it is denoted by Flux(X, µ, S). Our invariant is a minimax of the geometric flux, where one minimizes over all height functions and maximizes over the levels of the considered height function.

Definition 1. Assume that X is a vector field on S 3 or on a compact domain of R 3 that preserves a probability measure µ. Denote by φ X the flow of X. The trunkenness of X with respect to µ is

Tks(X, µ) := inf h height function max t∈[0,1] Flux(X, µ, h -1 (t)) = inf h height function max t∈[0,1] lim →0 1 µ(φ [0, ] X (h -1 (t))).
By a height function on S 3 we refer to a function with only two singular points and whose level sets are 2-dimensional spheres. On R 3 , the level sets of a height function all are topological planes.

From the definition, it is straightforward that the trunkenness of a vector field is invariant under diffeomorphisms that preserve the measure µ. More is true, the trunkenness is invariant under homeomorphisms that preserve µ.

Theorem A. Assume that X 1 and X 2 are vector fields on S 3 or on a compact domain of R 3 that preserve a probability measure µ and that there is a µ-preserving homeomorphism f that conjugates the flows of X 1 and X 2 . Then we have

Tks(X 1 , µ) = Tks(X 2 , µ).
What we do in this paper is to prove several properties of this new invariant. The first one is a continuity result that implies that the trunkenness of a vector field is an asymptotic invariant of order 1. For K a knot, we denote by Tk(K) its trunk (see Definition 1.1 below).

Theorem B. Suppose that (X n , µ n ) n∈N is a sequence of measure-preserving vector fields such that (X n ) n∈N converges to X and (µ n ) n∈N converges to µ in the weak- * sense. Then we have

lim n→∞ Tks(X n , µ n ) = Tks(X, µ).
In particular if X is ergodic with respect to µ then, for µ-almost every p, the limit

lim t→∞ 1 t Tk(k X (p, t))
exists and is equal to Tks(X, µ).

This continuity result then allows us to compute the trunkenness of some explicit vector fields on S 3 called Seifert flows. These computations in turn show that the trunkenness is not dictated by helicity, even in the case of ergodic vector fields, thus contrasting with most previously known knot-theoretical constructions.

Theorem C. There is no function f such that for every ergodic volume-preserving vector field X on S 3 one has Tks(X, µ) = f (Hel(X, µ)).

Finally we adress the question of what happens if for a non-singular vector field on S 3 there is a function that achieves the trunkenness, or in other words if the infimum in Definition 1 is a minimum.

Theorem D. Let X be a non-singular vector field on S 3 preserving the measure µ and h a height function such that Tks(X, µ) = max

t∈[0,1] Flux(X, µ, h -1 (t))
Then X has an unknotted periodic orbit.

One of the main motivations for constructing topological invariants of a vector field X is to find lower bound on the energies E p (X) := |X| p dµ. Indeed since a topological invariant yields a time-independent invariant of the physical system, an energy bound in term of a topological invariant will also be time-independent, although the energy may vary when the vector field is transported under (volume-preserving) diffeomorphisms. Such energy bounds exist for the helicity and for the asymptotic crossing number. We do not know whether the trunkenness bounds the energy.

The plan of the article is as follows. First we recall in Section 1 the definition of the trunk of a knot in order to make the definition for vector fields natural. Then we define the trunkenness of a measure-preserving vector field and prove Theorems A and B in Section 2. Using this result we compute of trunkenness of Seifert vector fields and prove Theorem C in Section 3. We prove Theorem D in Section 4. Finally in Section 5 we compute the trunkenness of some vector fields supported in the tubular neighborhood of a link.

Trunk of knots

For K a knot, we denote by K the set of all embeddings of K into R 3 . The standard height function on R 3 is the function h z : R 3 → R, (x, y, z) → z. Every level h -1 z (t) is a 2-dimensional plane. An embedding k ∈ K is said to be in Morse position with respect to h z if the restriction of h z to k is a Morse function. In this case there are only finitely many points at which k is tangent to a level of h z . Definition 1.1. Assume that k is an embedded knot in R 3 that is in Morse position with respect to h z . The trunk of the curve k relatively to h z is

tk hz (k) := max t∈R {k ∩ h -1 z (t)}.
The trunk of a knot K is then defined by Tk(K) := min

k∈K tk hz (k) = min k∈K max t∈R {k ∩ h -1 z (t)}.
The trunk of a knot was defined by M. Ozawa [START_REF] Makoto | Waist and trunk of knots[END_REF] and motivated by D. Gabai's definition of the waist of a knot [START_REF] David | Foliations and the topology of 3-manifolds III[END_REF]. between a horizontal level and the proposed embedding is 4. This number cannot be reduced under isotopy, hence the trunk of the trefoil is 4.

Example 1.2. A knot is trivial if and only if its trunk equals 2. Indeed the embedding as the boundary of a vertical disc shows that the trunk is less than or equal to 2, and every embedding in Morse position of the trivial knot has to intersect some horizontal plane in at least two points. Conversely, if the trunk of a knot is equal to 2, then it admits an embedding that intersects every horizontal plane in at most two points. The union of the segments that connect these pairs of points is a disc bounded by the knot, implying that the knot is trivial.

Example 1.3. For p, q in N, the torus knot T (p, q) can be realized as the closure of a braid with q strands, yielding Tk(T (p, q)) 2q. By symmetry one also gets Tk(T (p, q)) 2p. Actually, one can prove Tk(T (p, q)) = 2 min(p, q), see Remark 1.2 in [START_REF] Makoto | Waist and trunk of knots[END_REF].

Instead of fixing the function and changing the knot up to isotopy, one can fix the knot and change the function up to orientation preserving diffeomorphism. With this in mind, one defines a height function on R 3 as a function obtained by precomposing h z by a diffeomorphism, that is, a function of the form

h : R 3 → R (x, y, z) → h z (φ(x, y, z))
for φ an orientation preserving diffeomorphism of R 3 . In particular, a height function is a function whose levels are smooth planes. For K a knot and k a fixed embedding of

K in R 3 , one can then define tk h (k) := max t∈R {k ∩ h -1 (t)},
so that we have the alternative definition

(1)

Tk(K) = min h height function tk h (k) = min h height function max t∈R {k ∩ h -1 (t)}.

Trunkenness of measure preserving vector fields

We use the definition of Equation (1) to define the trunkenness of a vector field with respect to an invariant measure. The main question then concerns the analog of the number of intersection points of a surface with a curve when the curve is replaced by a vector field. A natural answer is the geometric flux. If X is a vector field that preserves a measure µ given by a volume element Ω, one can then consider the 2-form ι X Ω. For S a piece of oriented surface that is positively transverse to X, the integral S ι X Ω computes the instantaneous volume that crosses S. In other words, Figure 2. For S a piece of a surface (in red), the domain φ [0, ] (S) is roughly the part of the space located between S and φ (S). It is pinched around those points where the vector field is tangent to S. Its volume is • Flux(X, µ, S) at the first order.

by Fubini Theorem we have µ(φ

[0,t] (S)) = ( S ι X Ω) • t. On the other hand if S is negatively transverse to X we have µ(φ [0,t] (S)) = -( S ι X Ω) • t.
Therefore in this case, for any surface S, the instantaneous volume crossing S is given by S |ι X Ω|. Now if the measure µ is not given by integrating a volume form one cannot consider the above integral, but the quantity µ(φ [0,t] (S)) still makes sense for any piece of surface S (see Figure 2). Definition 2.1. For X a vector field that preserves a measure µ and for S a surface, the geometric flux of (X, µ

) through S is Flux(X, µ, S) := lim →0 1 µ(φ [0, ] (S)).
This definition generalizes the number of intersection points of a knot with a surface. Indeed one can see an embedding k of a knot K as a vector field with a particular invariant measure in the following way: consider a non-singular vector field X k that is tangent to k at every point and denote by φ t k the induced flow. Since k is closed, φ k is T k -periodic for some T k > 0. The Dirac linear measure associated to X k is defined by

µ k (A) = Leb({t ∈ [0, T k ], φ t k (x) ∈ A})
where A is a measurable set and x an arbitrary point on k. The measure µ k is X k -invariant and has total mass T k . In this setting, for S a surface that intersects k in finitely many points, a point p in the set k ∩ S has µ k -measure zero and thus cannot be detected by the measure. But by definition of µ k the set φ [0, ] k (p) is an arc of k of µ k -measure and since k ∩ S is made of finitely many points, for small enough, the set φ

[0, ] k (k ∩ S) has µ k -measure exactly • {k ∩ S}. In other words, one has {k ∩ S} = lim →0 1 µ k (φ [0, ] k (k ∩ S)). As µ k is concentrated on k, we thus have (2) {k ∩ S} = lim →0 1 µ k (φ [0, ] k (S)) = Flux(X k , µ k , S),
so the geometric flux indeed generalizes the number of intersection points.

We now mimic for vector fields the definition of the trunk of a knot. In order to have a welldefined maximum, in what follows we assume the vector fields are on a compact domain D 3 ⊂ R 3 or on the 3-sphere S 3 = R 3 ∪ ∞. In the later case, we define the standard height function

h 0 : S 3 → [0, 1] (x, y, z) → 1 - 1 1 + x 2 + y 2 + z 2 .
The levels h -1 0 (0) and h -1 0 (1) consist of the points (0, 0, 0) and ∞ respectively and every other level h -1 0 (t) is a 2-dimensional sphere of radius t/(1 -t) centered at the origin. A height function on S 3 is then a function obtained by precomposing by an orientation preserving diffeomorphism φ of S 3 , that is, a function of the form h : S 3 → [0, 1], (x, y, z) → h 0 (φ(x, y, z)).

Definition 2.2. Let X be a vector field whose flow preserves a measure µ on a compact domain of R 3 or on S 3 , and h a height function. We set

tks h (X, µ) := max t∈[0,1] Flux(X, µ, h -1 (t)) = max t∈[0,1] lim →0 1 µ(φ [0, ] (h -1 (t))).
The trunkenness of (X, µ) is defined as

Tks(X, µ) := inf h height function tks h (X, µ) = inf h height function max t∈[0,1] Flux(X, µ, h -1 (t)) = inf h height function max t∈[0,1] lim →0 1 µ(φ [0, ] (h -1 (t))).
Note that we can only consider an infimum instead of a minimum as in the case of knots. In Section 4 we prove that for non-singular vector fields, if the trunkenness is a minimum then the vector field possesses an unknotted periodic orbit.

If the invariant measure µ is given by the integration of a volume form Ω, we get the alternative definitions

tks h (X, Ω) = max t∈[0,1] h -1 (t)
|ι X Ω|, and Tks(X, Ω) = inf

h height function max t∈[0,1] h -1 (t) |ι X Ω|.
We now prove the invariance under homeomorphism.

Proof of Theorem A. Assume without loss of generality that there is a homeomorphism f such that f • φ t X1 = φ t X2 • f , with φ t Xi the flow of X i , and assume by contradiction that Tks(X 1 , µ) < Tks(X 2 , µ). Let 0 < δ = Tks(X 2 , µ) -Tks(X 1 , µ). Take h n to be a sequence of (differentiable) height functions such that lim n→∞ tks hn (X 1 , µ) = Tks(X 1 , µ).

Consider the continuous functions h n = h n •f -1 and for each n consider a differentiable function H n that is arbitrarily C 0 -close to h n . Hence, for each n, the level sets of H n and h n are arbitrarily close. Observe that we can assume that H n is a height function (by taking H n such that its level sets are all spheres or all planes, depending on the domain of the vector field). Thus we can choose H n so that, for every t ∈ [0, 1] and for small enough, we have

|µ(φ [0, ] X2 (H -1 n (t))) -µ(φ [0, ] X2 ( h -1 n (t)))| < δ 4 .
Fix n large enough so that tks hn (X 1 , µ) -Tks(X 1 , µ) < δ 4 , and for such n set T in [0, 1] so that tks Hn (X 2 , µ) = Flux(X 2 , µ, H -1 n (T )).

We have two possibilities:

(

1) If Flux(X 2 , µ, H -1 n (T )) ≥ Flux(X 2 , µ, h -1 n (T )), then for small enough µ(φ [0, ] X2 (H -1 n (T ))) ≤ δ 4 + µ(φ [0, ] X2 ( h -1 n (T ))). Hence Tks(X 2 , µ) ≤ tks Hn (X 2 , µ) = lim →0 1 µ(φ [0, ] X2 (H -1 n (T ))) ≤ δ 4 + lim →0 1 µ(φ [0, ] X2 ( h -1 n (T ))) = δ 4 + lim →0 1 µ(φ [0, ] X1 (h -1 n (T ))) ≤ δ 4 + tks hn (X 1 , µ) < δ 2 + Tks(X 1 , µ). Thus, δ = Tks(X 2 , µ) -Tks(X 1 , µ) < δ 2 , a contradiction proving Theorem A in this case. (2) If Flux(X 2 , µ, H -1 n (T )) < Flux(X 2 , µ, h -1 n (T )), then tks Hn (X 2 , µ) = Flux(X 2 , µ, H -1 n (T )) < Flux(X 2 , µ, h -1 n (T )) = Flux(X 1 , µ, h -1 n (T )) ≤ tks hn (X 1 , µ) < Tks(X 1 , µ) + δ 4 < Tks(X 2 , µ).
Thus, tks Hn (X 2 , µ) < Tks(X 2 , µ), a contradiction to the definition of the trunkenness. This finishes the proof of Theorem A.

Computing the trunkenness of a vector field is not easy in general. Considering a given height function gives an upper bound on the trunkenness, but lower bounds are harder to find. Theorem B is a continuity result that provides a useful tool.

Proof of Theorem B. We begin with the first part of the theorem, namely we prove that if (X n , µ n ) n∈N is a sequence of measure-preserving vector fields such that (X n ) n∈N converges to X and (µ n ) n∈N converges to µ in the weak- * sense, then we have lim n→∞ Tks(X n , µ n ) = Tks(X, µ).

Fix > 0. The convergence hypothesis implies that for every surface S, for every δ > 0 small and for n big enough, we have

(3) |µ n (φ [0,δ] Xn (S)) -µ(φ [0,δ]
X (S))| ≤ , where φ t X and φ t Xn denote the flows of the vector fields X and X n , respectively. Assume that Tks(X n , µ n ) does not converge to Tks(X, µ), then for any N ∈ N there exists n ≥ N such that |Tks(X n , µ n ) -Tks(X, µ)| > 3 . Fix n big enough satisfying the last inequality and (3). Then either Tks(X n , µ n ) < Tks(X, µ) or Tks(X, µ) < Tks(X n , µ n ).

Next we analyse these two cases and deduce a contradiction for each. If Tks(X, µ) < Tks(X n , µ n ), consider a sequence of height functions h k such that

lim k→∞ tks h k (X, µ) = Tks(X, µ).
Modulo extracting a subsequence we can assume that for all k we have that 0 ≤ tks h k (X, µ) -Tks(X, µ) ≤ . Observe that this difference is always positive by the definition of the trunkenness. Then

3 < Tks(X n , µ n ) -Tks(X, µ) = Tks(X n , µ n ) -tks h k (X, µ) + tks h k (X, µ) -Tks(X, µ), hence, 2 < Tks(X n , µ n ) -tks h k (X, µ) + tks h k (X n , µ n ) -tks h k (X n , µ n ) < Tks(X n , µ n ) -tks h k (X n , µ n ) + |tks h k (X n , µ n ) -tks h k (X, µ)|
For k fixed, since µ n tends to µ in the weak- * sense, we can assume that the term |tks h k (X n , µ n ) -tks h k (X, µ)| is smaller than by possibly taking n larger. Then we get 2 < Tks(X n , µ n ) -tks h k (X n , µ n ) + . We deduce + tks h k (X n , µ n ) < Tks(X n , µ n ), a contradiction to the definition since we have

+ tks h k (X n , µ n ) < Tks(X n , µ n ) ≤ tks h k (X n , µ n ).
The other case is similar. Assume now Tks(X n , µ n ) < Tks(X, µ) and for each n consider a sequence of height functions h n,k such that

lim k→∞ tks h n,k (X n , µ n ) = Tks(X n , µ n ).
As in the previous case, we assume that for all k we have 0

≤ tks h n,k (X n , µ n ) -Tks(X n , µ n ) ≤ . Then, for n large enough, 3 < Tks(X, µ) -Tks(X n , µ n ) = Tks(X, µ) -tks h n,k (X n , µ n ) + tks h n,k (X n , µ n ) -tks h n,k (X, µ) +tks h n,k (X, µ) -Tks(X n , µ n ) ≤ Tks(X, µ) -tks h n,k (X, µ) + + |tks h n,k (X, µ) -tks h n,k (X n , µ n )| ≤ Tks(X, µ) -tks h n,k (X, µ) + 2 .
We conclude that + tks h n,k (X, µ) < Tks(X, µ), a contradiction to the definition of Tks(X, µ). Thus, for n large enough, |Tks(X n , µ n ) -Tks(X, µ)| < 3 . This proves the first part of Theorem B.

The second part of Theorem B gives an asymptotic interpretation to the trunkenness of a vector field. Consider a sequence of knots (K n ) n∈N that support vector fields X n and denote by (t n ) n∈N the respective periods. From Equation (2) we get that for each n

{K n ∩ h -1 (t)} = lim →0 1 µ n (φ [0, ] n (h -1 (t))),
where µ n are the mesures supported by the knots and φ t n is the flow of X n . If we suppose that X n tends to X and that the normalized linear Dirac measures ( 1 tn µ n ) n∈N converge to µ in the weak- * sense, then we can reformulate the previous result into

lim n→∞ 1 t n Tk(K n ) = Tks(X, µ).
Now if X is ergodic with respect to µ then, for almost every p and for every sequence t n → ∞, the linear Dirac masses concentrated on the knots K n := k X (p, t n ) tend to µ in the weak- * sense. Recall that k X (p, t n ) is the knot obtained by following the orbit of p for a time t n concatenated with a geodesic from φ tn X (p) to p, as explained in the introduction. Then the convergence implies that lim t→∞ 1 t Tk(k X (p, t)) exists and is equal to Tks(X, µ). This proves the second part of Theorem B.

Independence of helicity

As mentioned in the introduction, the helicity is a well-known invariant of vector fields up to volume-preserving diffeomorphism. In this section, all vector fields are on the sphere S 3 and preserve a volume form, that we denote by Ω. For X such a vector field, Cartan's formula implies that ι X Ω is a closed 2-form, and since the ambient manifold is simply connected it is exact. We may then write ι X Ω = dα, for α some differential 1-form. The helicity of X is defined as

Hel(X) = S 3 α ∧ dα,
and does not depends on the choice of the primitive α.

As we recalled in the introduction most known asymptotic invariants are in fact proportional to a power of helicity [Arn73, GG01, Baa11, BM12]. The goal of this section it to prove that the trunkenness of a vector field is not a function of its helicity. In order to do so we compute the trunkenness and the helicity of a vector field that preserves the invariant tori of a Hopf fibration of S 3 .

Considering S 3 as the unit sphere

{(z 1 , z 2 ) ∈ C 2 , |z 1 | 2 +|z 2 | 2 = 1}
, the Seifert flow of slope (α, β) is the flow φ α,β given by φ t α,β (z 1 , z 2 ) := (z 1 e i2παt , z 2 e i2πβt ). It is generated by the vector field X α,β given by X α,β (z 1 , z 2 ) = (i2παz 1 , i2πβz 2 ). This flow preserves the standard volume form, that is, the volume form Ω Haar associated to the Haar measure of S 3 . The flow has two distinctive periodic orbits corresponding to z 1 = 0 and z 2 = 0 that are trivial knots in S 3 . The tori |z 1 /z 2 | = r for 0 < r < ∞ are invariant and the flow on each of them is the linear flow of slope α/β. If α/β is rational, put α/β = p/q with p, q ∈ N coprime. Then every orbit of φ α,β , different from the two trivial ones, is a torus knot of type T (p, q).

The helicity of φ α,β is equal to αβ. To compute it in the rational case (α, β) = (p, q) with p, q coprime, observe that all the orbits except two are periodic of period 1. The linking number of an arbitrary pair of such orbits is pq. Therefore the asymptotic linking number (also called asymptotic Hopf invariant) equals pq and, by Arnold's Theorem [START_REF] Vladimir | The asymptotic Hopf invariant and its applications[END_REF], so does the helicity. For the general case of (α, β) not necessarily rational, it is enough to use the continuity of the helicity, since X α,β can be approximated by a sequence of Seifert flows with rational slope.

Proposition 3.1. The trunkenness of the Seifert flow φ α,β with respect to the standard volume form Ω Haar is equal to 2 min(α, β).

Proof. Let us first prove Tks(X α,β , Ω Haar ) 2β. For this it is enough to exhibit a height function h that yields tks h (X α,β , Ω Haar ) = 2β. First define ∞ = (0, 1) and 0 = (0, -1) in S 3 ⊂ C 2 and take the stereographic projection to identify

{(z 1 , z 2 ) ∈ C 2 , |z 1 | 2 + |z 2 | 2 = 1} R 3 ∪ {∞}.
Take now as h the standard height function h 0 of R 3 ∪ {∞}. The spheres are centered at 0 ∈ R 3 that corresponds to the point (0, -1) ∈ S 3 ⊂ C 2 , hence the orbit z 1 = 0 intersects twice each level sphere h -1 0 (t). The middle sphere, S = h -1 0 (1/2), contains the other special orbit z 2 = 0 and is the only sphere that intersects all the orbits of φ α,β . Then the function t → h -1 0 (t) |ι X α,β Ω Haar | has a maximum for t = 1/2.

For computing S |ι X α,β Ω Haar |, we remark that the 2-sphere S has the orbit (e i2παt , 0) as an equator, that the flow is positively transverse to the northern hemisphere and negatively transverse to the southern hemisphere. Then the integral S |ι X α,β Ω Haar | is equal to twice the flux of X α,β through any disc bounded by the curve (e i2παt , 0). Consider the flat disc D in S 3 bounded by (e i2παt , 0). The first return time to D is constant and equal to 1/β, so the flux multiplied by 1/β gives the total volume of S 3 , that is 1. Therefore Flux(X α,β , Ω Haar , D) is equal to β, and we obtain tks h0 (X α,β , Ω Haar ) = S |ι X α,β Ω Haar | = 2β. By symmetry, we then have Tks(X α,β , Ω Haar ) 2 min(α, β).

For proving the converse inequality Tks(X α,β , Ω Haar ) 2 min(α, β), we approximate X α,β by a sequence (X pn/rn,qn/rn ) n∈N , where p n , q n , r n are integer numbers. Theorem B yields Tks(X α,β , Ω Haar ) = lim n→∞ Tks(X pn/rn,qn/rn , Ω Haar ).

As the trunkenness is an order-1 invariant (it is multiplied by λ if the vector field is multiplied by λ), we only have to prove Tks(X p,q , Ω Haar ) = 2 min(p, q) for p, q two coprime natural numbers.

Since every orbit of X p,q is periodic, we can consider a sequence of (K n ) n∈N of collections of periodic orbits whose induced linear Dirac measures converge to Ω Haar . We take K n to be an n-component link all of whose components are torus knots T (p, q). Actually K n is a cabling with n strands on T (p, q), so by Zupan's theorem [START_REF] Alexander | A lower bound on the width of satellite knots[END_REF], the trunk of K n is 2n min(p, q). Since the period of each component of K n is 1, the total length of K n is n, and we get Tks(X p,q , Ω Haar ) 2 min(p, q). Proof of Theorem C. The previous computations show that for a Seifert flow X α,β on S 3 we have Hel(X α,β , Ω Haar ) = αβ and Tks(X α,β , Ω Haar ) = 2 min(α, β). There is no real function g such that min(α, β) = g(αβ), so there is no function g such that Tks(X α,β , Ω Haar ) = g(Hel(X α,β , Ω Haar )).

However the Seifert flows are not ergodic with respect to Ω Haar . Indeed, the foliation of S 3 by invariant tori is invariant, so that it is easy to construct an invariant set with arbitrary mesure. Still, a theorem of Katok [START_REF] Anatole | Ergodic perturbations of degenerate integrable Hamiltonian systems[END_REF] states that every volume-preserving vector field can be perturbed (in the C 1 -topology) into an ergodic one. Starting from X 1,8 and X 2,4 , we obtain two ergodic volume-preserving vector fields X 1,8 and X 2,4 . By continuity, their trunknesses are close to 2 and 4 respectively, while their helicities are close to 8. At the expense of multiplying the X 1,8 and X 2,4 by a constant, we can assume that their helicities are exactly 8. However their trunknesses are still close to 2 and 4, hence different.

The formula Tks(X α,β , Ω Haar ) = 2 min(α, β) is also interesting to compare with Kudryavtseva's and Encisco-Peralata-Salas-Torres de Lizaur's theorems: the function (α, β) → 2 min(α, β) is continuous but not differentiable, so that trunkenness is a continuous vector field invariant, but it is not integral regular in the sense of [START_REF] Elena | Helicity is the only invariant of incompressible flows whose derivative is continuous in C 1 -topology[END_REF][START_REF] Alberto | Helicity is the only integral invariant of volume-preserving transformations[END_REF].

Trunkenness and the existence of periodic orbits

In this section we adress the question of what happens when the infimum in the definition of the trunkenness is a minimum, for non-singular vector fields on S 3 with an invariant mesure µ. We deduce that the vector field must posses an unknotted periodic orbit by finding a periodic orbit tangent to a level of the function. The proof of Theorem D in particular implies that there is height function h such that

tks h (X, µ) = Flux(X, µ, h -1 (t max ))
for some (not necessarily unique) t max ∈ [0, 1] and such that all the connected components of S = h -1 (t max ) along which X is tangent to S are bounded by periodic orbits of X. The existence of vector fields on S 3 without periodic orbits [START_REF] Krystyna | A smooth counterexample to the Seifert conjecture[END_REF] implies that there are vector fields for which we cannot consider a minimum to define the trunkenness.

Proof of Theorem D. For f a height function we define

F f : [0, 1] → R t → Flux(X, µ, f -1 (t))
Set h the height function such that Tks(X, µ) = tks h (X, µ) and let t max ∈ [0, 1] be a maximum of

F h , hence Tks(X, µ) = tks h (X, µ) = lim →0 1 µ(φ [0, ] X (h -1 (t max ))).
Denote by S the level set h -1 (t max ). Observe that the value t max is not necessarily unique.

We assume that S 3 and S are oriented and thus we can distinguish three subsets of S (see Figure 3):

• S t the closed set along wich X is tangent to S;

• S + the open set along wich X is positively transverse to S;

• S -the open set along wich X is negatively transverse to S. We claim that none of these sets is empty. First assume that S + is empty, then S = S t ∪ S -. Observe that S 3 \ S has two connected components that are diffeomorphic to open 3-dimensional balls. Hence there is one of these connected components, denote its closure by D, that is invariant under the diffeomorphism of S 3 defined by φ t X for any t > 0, and since X is non-singular there exists t 0 > 0 such that Φ := φ t0 X has no fixed points. Thus Φ maps D to D, and D is the closed 3-dimensional disc, hence by Brouwer fixed point theorem Φ has a fixed point, a contradiction. Thus S + is non-empty, and the same argument proves that S -is non-empty. Now S + and S -are open subsets of S and have empty intersection, hence S t = S \ (S + ∪ S -) is not empty.

S - S + S t Figure 3.
The decomposition of the level S of the height function into S + where X is positively transverse to S (orange), S -where X is negatively transverse (black), and S t where X is tangent to T (white). A piece of orbit of X that intersects S four times is also shown.

Decompose B := ∂S t as B = B + ∪ B -, where B + and B -are in the boundary of S + and S - respectively. Observe that there is at least one connected component of S t whose boundary has both positive and negative parts.

Assume that X has no periodic orbits, then B is made of circles and each one of these circles is tranverse to X in at least one point. Lemma 4.1. If a point p in S t is such that either its positive or negative orbit is contained in S, then X has an unknotted periodic orbit.

Proof. Assume without loss of generality that the positive orbit of p is contained in S, then it limits to some invariant set of X contained in S. By Poincaré-Bendixson Theorem such a set has to be a periodic orbit of X. Since the periodic orbit is contained in the sphere S it has to be unknotted.

Then under the assumption that X has no unknotted periodic orbits, we have that for every p ∈ S t the positive and negative orbit of p have to leave S at some point. Consider now two points p, q ∈ B that are in the same orbit and such that the orbit segment connecting them is contained in S t . We can thus assume that there exists τ ≥ 0 such that φ τ X (p) = q and φ s X (p) ∈ S t for all 0 ≤ s ≤ τ . Observe that by allowing τ to be zero, we consider the case p = q.

Take > 0 and consider the orbit segment

O = {φ s X (p) | -< s < τ + }.
The flowbox Theorem implies that there is a neighborhood U of O and a diffeomorphism ϕ : B(1) × (-, τ + ) → U so that the flowlines are the image under ϕ of the vertical segments {•} × (-, τ + ). Here B(1) denotes the 2-dimensional open disc of radius 1. Proposition 4.2. There exists p, q ∈ B as above such that p ∈ B -and q ∈ B + or p ∈ B + and q ∈ B -.

The idea of the proof is that if for every pair of points p, q as above that are both in B + or B -, we can change the function h for another height function h 1 such that tks h (X, µ) = tks h1 (X, µ) and such that the level of h 1 realizing the trunkenness has no tangent part that separates S + from S -, which is impossible.

U V V 0 S + S t S -
Proof. Consider a pair of points p, q ∈ B + as above (see Figure 4). Let V 0 be the neighborhood of O defined as ϕ(D(1/3) × (-, τ + )), where D(1/3) is the 2-dimensional closed disc of radius 1/3 and V = ϕ(B(2/3) × (-, τ + )). Hence V 0 ⊂ V ⊂ U . We will deform the levels of h intersecting V without changing the trunkenness. The image of S by ϕ -1 defines a surface Σ that is positively transverse to the vertical direction in B(1) × (-, 0) and B(1) × (τ, τ + ), as in Figure 5 left. If we project Σ to B(1) × {0}, the flux through Σ and through the image of Σ under the projection are the same. Thus if we change Σ (or any surface) for another surface whose projection, counted with multiplicities and signs, is the same as the one of Σ, the flux remains constant.

Inside D(1/3) × (-, τ + ) we change the images under ϕ -1 of the levels of h to obtain a family of surfaces that are always positively transverse to X and are C ∞ -close to the original ones, in such a way that the projection to B(1) × {0} is preserved (as in Figure 5 right). Using the neighborhood V , we can paste the deformed surfaces of V 0 with the original surfaces in U \ V .

Using ϕ this deformation can be pushed forward to the manifold so that the level surfaces of h that intersected V are modified and X is positively transverse to the surfaces inside V 0 . Let S be the surface obtained from S after the deformation and h be a height function whose level sets correspond to the deformed surfaces. Then tks h (X, µ) = tks h (X, µ), since Flux(X, µ, S) = Flux(X, µ, S), and for every surface intersection V the corresponding equation holds. Clearly, the same proof works if p, q ∈ B -.

Observe that the level S of h realizing the trunkenness coincides with S outside V . Moreover, the tangent part S t of S is strictly smaller than S t : the connected component A of S t containing p and q got transformed into A \ (V 0 ∩ A). In other words there is now a strip accross A.

Assume now that for any couple of points p, q ∈ B that is joined by an orbit segment contained in S t we have either p, q ∈ B + or p, q ∈ B -. The sets S t and B are compact, we can thus find a finite number of pairs of points p, q ∈ ∂B such that after deforming the levels h as above along each of the corresponding orbit segments, we obtain a height function h 1 such that tks h (X, µ) = tks h1 (X, µ) = Flux(X, µ, S 1 ), where S 1 is the level of h 1 realizing the trunkenness and such that S t 1 is formed by discs. Hence none of the connected components of S t 1 separates the positively tranverse set S + 1 from the negatively transverse set S - 1 , a contradiction. Consider now a pair of points p, q as in Proposition 4.2 and assume that p ∈ B -and q ∈ B + . We now construct a height function h 2 such that tks h (X, µ) > tks h2 (X, µ), a contradiction that implies that X has unknotted periodic orbits.

Consider the function F h and let M h be the set of t ∈ [0, 1] realizing the trunkenness, that is

M h = {t ∈ [0, 1] | F h (t) = tks h (X, µ)}.
Observe that M h is closed. We distinguish two cases:

(1) all the elements of M h are isolated;

(2) there exist a closed interval I max contained in M h . We start with the first case. Take t max ∈ M h and set S = h -1 (t max ). To obtain h 2 we will deform the levels near S, if M h has more than one element the following deformation has to be done near each correponding level set. Thus we assume without loss of generality that t max is the only element of M h .

Let O, ϕ, U , V and V 0 be defined as above

. Let [a, b] ⊂ [0, 1] be such that t max ∈ [a, b] and if h -1 (t) ∩ V 0 = ∅ then t ∈ [a, b]. Shrinking V if necessary, we can assume that ϕ(B(2/3) × (-, τ + ) is contained in h -1 ([a, b]) and that for any t ∈ [0, 1] \ [a, b] we have Flux(X, µ, h -1 (t)) < tks h (X, µ) -δ,
for some δ > 0. We assume also that ϕ(B(2/3) ×

[0, τ ]) is contained in h -1 ([a, b]).
Consider now the surface Σ obtained as the image under ϕ -1 of S in U . Then either Σ is positively transverse to the vertical direction in B(1) × (-, 0) and negatively transverse to the vertical direction in B(1) × (τ, τ + ), or it is negatively transverse to the vertical direction in B(1) × (-, 0) and positively transverse to the vertical direction in B(1) × (τ, τ + ). We assume without loss of generality that Σ is negatively transverse to the vertical direction in B(1) × (-, 0) and positively transverse to the vertical direction in B(1) × (τ, τ + ), as represented in Figure 6.

We want to deform the levels of h intersecting V 0 in such a way that their trace under the projection to B(1) × {0} is reduced.

Shrinking V if necessary, we can assume that the level sets are:

• tangent or negatively transverse to X in ϕ(B(2/3) × (-, 0)),

• tangent or positively transverse to X in ϕ(B(2/3) × (τ, τ + )).

Consider the circle

C = Σ ∩ ∂(D(1/3) × (-, τ + )). Since ∂(D(1/3) × (-, τ + )
) is topologicaly a sphere, C divides it into two discs. Let W be the disc that is entirely tangent to the vertical direction. There is a continuous deformation from Σ to the continuous surface Σ that coincides with Σ outside D(1/3) × (-, τ + ) and coincides with W in this set. Observe that the projection of Σ to B(1) × {0} is a proper subset of the projection of Σ. We can now approximate Σ in B(2/3) × (-, τ + ) with a smooth surface Σ 1 , that can be obtained from Σ by a continuous deformation that is the identity near the boundary of B(2/3) × (-, τ + ). Apply this deformation to nearby levels by pushing (to the right as in Figure 6) all the level sets intersecting B(1/3) × (0, τ ) with a continuous map that is the identity near the boundary of B(2/3) × (-, τ + ). By construction the projection to B(1) × {0} of the image under ϕ -1 of any level set intersecting V is a subset of the original one and for levels sufficiently near h -1 (t max ) it is a proper subset. Thus the flux of X gets reduced on any of the deformed surfaces.

As before, we use ϕ to push forward the deformation to S 3 and obtain the levels of a new height function h 2 such that tks h2 (X, µ) < tks h (X, µ).

The proof of the case where M h has only isolated elements ends here and we are left with the case where the set M h contains an interval I max = [a, b]. Take t max = a and use the previous procedure to obtain a new height function h 2 such that the interval I max is reduced to I 2 = [a 2 , b] for some a 2 > a. This process can be applied as long as the boundary of the tangent part of h -1 (t) for any t ∈ [a, b] is not composed by periodic orbits of X. Recursively either we find an unknotted periodic orbit or we obtain a height function f such that tks f (X, µ) < tks h (X, µ) as needed.

The proof of Theorem D gives the following result.

Corollary 4.3. Let X be a non-singular vector field on S 3 , µ an invariant measure and h a height function such that Tks(X, µ) = tks h (X, µ).

Then there is a height function h 1 such that:

• tks h (X, µ) = tks h1 (X, µ); • for every s ∈ [0, 1] such that tks h1 (X, µ) = Flux(X, µ, h 1 -1 (s)) and for every point p in the boundary of the tangent part of S = h -1 1 (s), either p belongs to a periodic orbit or its orbit limits to a periodic orbit. In both cases the periodic orbit is contained in S and is thus unknotted. to the canonical meridian discs. The top one has all its orbits horizontal, hence its trunkenness is 0 while the bottom one has linked orbits. Its trunkenness is at most 2 • Flux(X, Ω, D) but it might be smaller. On the right a vector field supported on a link L both of which components are figure-eight knots. If the flux of the vector field through the canonical meridian discs of both components is equal to F , then the trunkenness of this vector field is F • Tk(L) = 8F by Proposition 5.1.

Trunkenness of knotted tubes

In this section we compute the trunkenness with respect to a volume form of some vector fields supported in a tubular neighborhood of a link or knot. Our statement is reminiscent from Zupan's theorem [START_REF] Alexander | A lower bound on the width of satellite knots[END_REF] concerning the trunk of the cable of a knot. Recall that for a divergence-free vector field supported on a tube, the fluxes through all meridian discs are equal.

Proposition 5.1. (see Figure 7 right) Suppose that X is a Ω-preserving vector field supported on tubes T 1 , . . . , T j that are tubular neighborhoods of knots k 1 , . . . , k j , each knot k i being non-trivial, and such that X is transverse to all canonical meridian discs and the flux of X through each of them is constant equal to some F = 0, then we have

Tks(X, Ω) = F • Tk(k 1 ∪ • • • ∪ k j ).
Note that the trunkenness in this case is independent from the dynamics of X inside each tube, and in particular of the first-return map on a meridian disc, exactly like the trunk of a cable link is independent of the twist number of the cabling. However the restriction that all the knots k i are non-trivial is necessary. Indeed the trunkenness of a vector field supported on a tube that forms a trivial knot may be smaller than 2F . For example take an unknotted solid torus in R 3 that supports a vector field obtained by the suspension of the identity (with no extra twist) and assume that the vector field is zero outside the solid torus (see Figure 7 top left). The trunkenness of such vector field is zero: it is enough to consider a height function whose levels are always tangent to the vector field. Observe that the levels of such a height function intersect the solid torus along circles or annuli. As this example shows, the trunkenness of a vector field supported in the neighborhood of a link with at least one unknotted component seems hard to determine (see Figure 7).

Proof. Let L = k 1 ∪ k 2 ∪ . . . ∪ k n ⊂ S 3 be a link all of whose components are non-trivial knots. Let h be a height function such that Tk(L) = tk h (L). Observe that we can assume that h exists since for a knot or link the trunk is defined by a minimum (see Definition 1.1). The trunk of L is then realized by one or more level sets of h, that is, there exist t 1 , t 2 , . . . , t k such that for any 1 ≤ i ≤ k there is a critical value c i of h| L so that t i < c i < t i+1 and Tk(L) = {L ∩ h -1 (t i )}. Let m j (t i ) = {k j ∩ h -1 (t i )}, for 1 ≤ j ≤ n and 1 ≤ i ≤ k.

Figure 1 .

 1 Figure 1. The trunk of the trefoil knot: the maximal number of intersection points

Figure 4 .

 4 Figure 4. A neighborhood U of a piece of orbit O of X that connects two points p, q in B + . The intersection of ∂U with S is bold and green.

Figure 5 .

 5 Figure5. Modifying the levels of h in U around the arc pq in the case p, q ∈ B + .

Figure 6 .

 6 Figure 6. Modifying the levels of h in U around the arc pq in the case p ∈ B + , q ∈ B -.

Figure 7 .

 7 Figure 7. On the left, two vector fields supported on the unlink and both transverse

Instituto de Matemáticas, Universidad Nacional Autnoma de México, Ciudad Universitaria, 04510 Ciudad de México, México rechtman@im.unam.mx http://www.matem.unam.mx/fsd/rechtman

Acknowledgements. We thank Michel Boileau who suggested to study the trunk instead of the genus for vector fields during a visit of P.D. to Toulouse in 2013.

For any 0 < < 1 there is an Ω-preserving diffeomorphism f of S 3 that makes the solid tori T j longer and of radius r j for any 1 ≤ j ≤ n, with r j the radius of the tubular neighborhood T j . Let X be the vector field obtained from X via f , then Tks(X, Ω) = Tks(X , Ω) for every .

Assume that there is a height function g and an index i such that tks g (X, Ω) < F • n j=1 m j (t i ). Consider the height function g = g • f -1 , then Theorem A implies tks g (X, Ω) = tks g (X , Ω) and

We claim that tk g (L) < n j=1 m j (t i ) = Tk(L) holds, which is absurd. Assume first that g restricted to the core of the tori f (T j ) is a Morse function. The core of the tori form the link L thus tk g (L) ≥ Tk(L) in one hand. But since

for some T ∈ [0, 1], taking arbitrarily small tells us that the number of discs in the intersection of g -1 (T ) and the tori is smaller than Tk(L). In other words we conclude that tk g (L) < Tk(L), which is impossible. We are left with the case where g is not a Morse function when restricted to L, seen as the core of the tori f (T j ). The following lemma finishes the proof of the proposition.

Lemma 5.2. In the previous context, there exists another height function g so that:

• g | L is a Morse function;

• tks g (X , Ω) = tks g (X , Ω).

Proof. Let S = g -1 (t) be a level that is tangent to at least one of the components of L and let k 1 be one of the tangent components. We will modify g in a neighborhood of k 1 , this modification taking place near the part of k 1 that is tangent to S.

Observe that S is tangent to a closed subinterval of k 1 , since if it were tangent to all of k 1 then k 1 would be the trivial knot, in contrary to our assumption on L. For δ < r 1 , consider N (δ, k 1 ) the δ-tubular neighborhood of k 1 that is contained in f (T 1 ). For δ sufficiently small, the intersection S ∩ N (δ, k 1 ) is composed by a finite number of discs. Let D 1 be one of these discs, then ∂D 1 is included in ∂N (δ, k 1 ) and is either a contractible circle in ∂N (δ, k 1 ) or not.

If ∂D 1 is contractible in ∂N (δ, k 1 ), then we can locally deform the levels of g near D 1 , to obtain a new height function g such that:

• the disc D 1 obtained from D 1 is tangent to k 1 at one point;

• near D 1 all levels are transverse to k 1 ;

• tks g (X , Ω) = tks g (X , Ω). The support of the deformation is contained in N (δ, k 1 ). Performing this deformation near any disc in S ∩ N (δ, k 1 ) whose boundary is contractible in ∂N (δ, k 1 ), we can assume now that the discs of S that are tangent to k 1 have non-contractible boundary in ∂N (δ, k 1 ). Thus these discs are homologous to meriodional discs D 2 × {θ} of T 1 . Again, we can locally deform the levels, so that this discs are transverse to k 1 . The deformed height function g is Morse when restricted to L.

Thus g is Morse when restricted to k 1 . Repeating this process if necessary, we obtain a that g is Morse when restricted to L.