Frédéric Kuznik
email: frederic.kuznik@insa-lyon.fr

Joseph Virgone

Experimental Assessment of a Phase Change Material for Wall Building Use

Keywords: phase change material, wallboard, energy storage, experimental investigation, temperature fluctuation

The thermal performances of a PCM copolymer composite wallboard has been experimentally investigated in a full scale test room. The test cell is totally controlled so that a typical day can be repeated (temperature and solar radiative flux). Effects of the PCM are investigated comparing the results obtained with and without composite wallboards for three cases: a summer day, a winter day and a mid-season day. The results show that: (1) for all the cases tested, the decrement factor varies between 0.73 and 0.78 which is quite interesting for use in buildings and particularly for renovation; (2) the air temperature in the room with PCM lowers up to 4.2 • C, the comfort enhancement is more important if the surface temperatures are also considered; (3) the PCM wallboards enhance the natural convection in the room and then there is no thermal stratification contrary to the room without composite; (4) the numerical experiments are fully described and can then be used to evaluate PCM numerical modeling.

Introduction

Nowadays, thermal energy storage systems are essential for reducing dependency on fossil fuels and then contributing to a more efficient environmentally benign energy use [START_REF] Dincer | Thermal Energy Storage -Systems and Applications[END_REF]. As demand in thermal comfort of buildings rise increasingly, the energy consumption is correspondingly increasing. For example, in France, the energy consumption of buildings has increased by 30% the last 30 years. Housing and tertiary buildings are responsible for the consumption of approximatively 46% of all energies and approximatively 19% of the total CO 2 emissions [START_REF]Climate Plan 2004: let's act together to challenge of climate change[END_REF].

Thermal energy storage can be accomplished either by using sensible heat storage or latent heat storage. Sensible heat storage has been used for centuries by builders to store/release passively thermal energy, but a much larger volume of material is required to store the same amount of energy in comparison to latent heat storage. The principle of the phase change material (PCM) use is simple. As the temperature increases, the material change phase from solid to liquid. The reaction being endothermic, the PCM absorbs heat. Similarly, when the temperature decreases, the material changes phase from liquid to solid. The reaction being exothermic, the PCM desorbs heat.

The main disadvantage of light weight buildings is their low thermal mass. ing, solar heat or heating loads. Using PCM material in such building walls can decrease the temperature fluctuations, particularly in case of solar radiations loads. It is then a potential method for reducing energy consumption in passively designed buildings. This tendency is confirmed by numerous papers available in the literature during the last 20 years. For a review, see in [START_REF] Tyagi | PCM Thermal Storage in Buildings: A State of Art[END_REF], [START_REF] Khudhair | A review on energy conservation in building applications with thermal storage by latent heat using phase change materials[END_REF] and [START_REF] Zhang | Application of latent heat thermal energy storage in buildings: State-of-the-art and outlook[END_REF].

When selecting a PCM, the average room temperature should be close to the melting/freezing range of the material. Moreover, the day temperature and solar radiations fluctuations should allow the material phase change. Then many factors influence the choice of the PCM: weather, building structure and thermophysical properties [START_REF] Darkwa | Phase-change drywalls in a passivesolar building[END_REF],..That's why experiments must be carried out to effectively assess the use of PCM.

In this study, three cases are tested, depending on the simulated season day.

The PCM wallboard allows to reduce the overheating due mainly to solar radiations during the summer time. Then, the first case tested is a summer day. During the winter, the PCM composite included in walls stores surplus of energy during the day and releases it during the night. This feature is then examined simulating a winter day. The mid-season is the longest period of a year; the third case tested is then a mid-season day.

The first part of this paper deals with a bibliographical review of the PCM used in building walls (part 2). The experimental set-up is presented in the part 3 of this article. The PCM composite, the experimental test cell and the experimental protocol are fully detailed. In the third part, the experimental results are analyzed in terms of comparative thermal performances for the cases with and without PCM wallboard. Finally, the part 5 concerns the main conclusions of this study.

Bibliographical review

The PCM incorporated in building walls are used to enhance the storage capacity of a light weight structure. Usually, the PCM is incorporated either in gypsum or in concrete. In [START_REF] Hawes | Latent heat storage in building materials[END_REF], dodecanol is incorporated in pumice concrete block. Even if the results are interesting for energy storage, the latent heat of the composite is relatively low. That's why the most interesting possibility in building application is the impregnation of PCM into porous material such as plasterboard.

Fatty acids and organic phase change materials were used in [START_REF] Feldman | Energy storage composite with an organic phase change material[END_REF], [START_REF] Feldman | Fatty acids and their mixtures as phase change materials for themal energy storage[END_REF] and [START_REF] Feldman | Obtaining an energy storing building material by direct incorporation of an organic phase change material in gypsum wallboard[END_REF].

Various building materials were used to evaluate their absorption properties and the resulting composites. The gypsum can absorb up to 25%wt of PCM.

The results show that gypsum-PCM composite can be used to reduce overheating.

Gypsum wallboard has also been studied, numerically and experimentally in a full scale test room, in [START_REF] Athienitis | Investigation of the thermal performance of a passive solar test-room with wall latent heat storage[END_REF]. Their numerical results are in good agreement with experimental data. moreover, the PCM used shows a decrease in overheating and a decrease of energy consumption.

PCM can be microencapsulated for being included in building materials. However, thermal performances of the composite highly depend on the microencapsulation process [START_REF] Hawlader | Microencapsulated PCM thermalenergy storage system[END_REF]. In [START_REF] Schossig | Micro-encapsulated phase-change materials integrated into construction materials[END_REF], micro-encapsulated PCM is integrated into gypsum wallboard. full-size tests show that such composite lowers the tem-perature by up to 4 • C and significantly reduces the number of hours with temperature above 28 • C.

The main problem with gypsum is the quantity of PCM that can be incorporated: the maximum weight ratio is about 30%wt. To overcome this problem, [START_REF] Ahmad | Thermal testing and numerical investigation of a prototype cell using light wallboard coupling vacuum isolation panels and phase change material[END_REF] filled a PVC panel with polyethylene glycol phase change material. The results show that the small test cell temperature amplitude decreases up to 20 • C.

On the whole, the use of PCM in classical building construction material is limited by the possibility to incorporate an important quantity of material (up to 30%wt in gypsum). The composite material tested in this paper constituted of 60%wt of microencapsulated paraffin within a copolymer. Moreover, the surface mass of the PCM composite is 4.5kg/m 2 which is quite lower than the plaster board value i.e. 8.1kg/m 2 for a 10mm thickness. Then, it is an interesting candidate for enhancing thermal storage capacity of light weight buildings, especially for renovation.

Description of the experimental set-up

The aim of this article is to evaluate a composite PCM using a comparative study. Thus, two series of experiments are led, with and without PCM. The section 3.1 is dedicated to a physical presentation of the composite PCM wallboard. The section 3.2 deals with the description of the full-scale test room MINIBAT. Finally, in section 3.3 the experimental protocol is explained.

Phase change material tested

The product tested, EN ERGAIN has been achieved by the Dupont de Nemours Society and is constituted of 60% of microencapsulated paraffin within a copolymer. The final form of the composite PCM (see figure 1) is a flexible sheet of 5mm thickness which density is about 900kg.m -3 . The thickness of the PCM is the result of a commercial compromise and allows 77% of optimal efficiency obtained with 1cm thickness [START_REF] Kuznik | Optimization of a phase change material wallboard for building use[END_REF].

The thermal conductivity has been measured using guarded hot-plate apparatus [START_REF] Astm | Standard Test Method for Steady-State Heat Flux Measurements and Thermal Transmission Properties by Means of the Guarded-Hot-Plate Apparatus[END_REF]. The thermal conductivity is 0.22W.m -1 .K -1 in liquid phase and decreases to about 0.18W.m -1 .K -1 in solid phase.

The composite PCM enthalpy has been measured using a differential scanning calorimeter as illustrated in figure 2; the heating and cooling rate being From the DSC curves, melting and freezing temperatures are 13.6 • C and 23.5 • C respectively. The measured latent heats of melting and freezing are respectively 107.5J/g and 104.5J/g ;and 72.4J/g and 71J/g for the range

[5 • C; 30 • C].
The table 1 reports thermal energy storage data from literature concerning building materials. From this table, the composite PCM described in this article has an important potential of thermal energy storage in building walls. This particularity is due to the possibility to incorporate much more PCM in the polymeric material than in a traditional construction material, for example: ⋆ 35wt% of paraffin in gypsum -see [START_REF] Scalat | Full scale thermal testing of latent heat storage in wallboard[END_REF], ⋆ 5.6wt% of butyl sterate PCM in concrete block; 8.6wt% of paraffine in concrete block -see [START_REF] Lee | Control aspects of latent heat storage and recovery in concrete[END_REF]. The test cell, of volume 3.10x3.10x2.50m 3 , is bounded on five sides by air volumes regulated at a constant temperature (the thermal guard). The sixth face is a glazed facade which isolates the test cell from a climatic chamber.

A solar simulator completes the experimental set-up and allows to generate short-wave radiations in the test cell.

The thermal guard

The air in the thermal guard is controlled by the means of an air treatment system. Air diffusers are installed in the upper part of the thermal guard air whereas the air is extracted in the lower part. The configuration of the air distribution allows an acceptable air mixing inside the thermal guard. The air temperature inside the zone is entirely controlled with an accuracy of ±0.5 • C.

The temperature of the thermal guard is set to 20.5 • C in our experiments to simulate adjacent rooms.

The climatic chamber

The climatic chamber temperature can vary between -10 • C and 40 • C and can be dynamically controlled so that any temperature evolution can be generated.

Fans are also used to obtain a homogeneous temperature in the chamber.

The solar simulator

In order to have a light source which reproduces best the solar effects, 1000W CSI lamp is selected (gas-discharge lamps with metal halide). The figure 4 presents the comparison between CSI lamp and solar spectrum [START_REF] Allard | Thermal Experiments of Full-scale Dwelling Cells in Artificial Climatic Conditions[END_REF].

12 spotlights are placed on 3 horizontal lines (see figure 5

The test cell

The compositions of the walls are described in the table 2. The physical characteristics of the materials used are summarized in table 3. All the opaque walls are covered with the same coating: a gray diffuse paint which physical properties are a solar absorptivity α s = 0.67 and a global emissivity ε g = 0.95.

The measurement devices

All the faces temperatures (internal and external) are measured using thermocouples of resolution ±0.4 • C, each face being equipped with 9 thermocouples.

The temperatures of the climatic chamber and the different parts of the thermal guard are measured using Pt100 probes with an accuracy of ±0.3 • C. The air temperature of the test cell is measured using radiation shielded Pt100 probes: the first one is positioned in the middle of the room at a height of 85cm ; the second one is at a height of 170cm. The time evolution of the vertical radiative flux density on the glazed facade is measured using a pyranometer.

The acquisition of the various parameters is done by the means of a multiplexermultimeter connected to a PC. The control of the whole of the apparatuses, except climatic control, is made using software LABVIEW. The time step chosen between two series of measurement is 10mn and the duration of each test is three days. The results presented in the paper concern the two last days, the first being used to erase initial conditions effects.

Experimental protocol

Three of the test cell walls are modified to include, or not, the composite PCM described in part 3.1. The locations of these walls are shown in figure 5: these are north, east and west walls. The configurations of the walls with and without PCM are described in figure 6.

1 U = i e i λ i (1)
with e i the thickness of the layer i and λ i its thermal conductivity.

Due to composite PCM properties, the U-value of the two walls tested are very close, with a value of about U = 0.59W/m 2 K. This feature is important to neglect the PCM composite insulation property in the problem.

The test presented in this article are of three types:

• a summer day case, for which the temperature of the climatic chamber varies between 15 • C and 30 • C. In this case, there is a night cooling. For that purpose, a ventilation is switched-on between [6h -18h] and [30h -42h] (according to figure 7 it corresponds to the T cl < 22.5 • C -the time scale of the figures does not correspond to a day time scale), the airflow rate being 92m 3 /h (ie. 3.8ACH) and the air coming from the climatic chamber. Such night ventilation is suitable to improve the PCM storage/release effects [START_REF] Zhou | Numerical analysis of effect os shapestabilized phase change material plates in a building combined with night ventilation[END_REF].

• a mid-season day case, for which the temperature of the climatic chamber varies between 10 • C and 18 • C.

• a winter day case, for which the temperature of the climatic chamber varies between 5 • C and 15 • C. A 1500W heating system has been placed into the room with a regulation at 20 • C (the heating system works when the temperature in the room is below 20

• C).
For all the cases tested, the solar radiative fluxes are the same. The climatic chamber temperature T cl and the radiative fluxes (E) are presented figure 7.

Obviously, these conditions are the same for the cases with and without PCM materials. We can notice the good repeatability, for the chosen period of 24h, of the controls of the climatic chamber temperature and lighting of the projectors.

In conclusion, the experimental methodology allows us to obtain a complete In this paper, the PCM effect on indoor air temperature is investigated using the decrement factor f . The decrement factor is the ratio between the amplitude of the indoor air temperature in the cell with PCM and the amplitude of the reference test cell air temperature (ie. with regular wallboards). For the summer case, the decrement factor is f = 0.79.

An interesting observation concerns the differences between temperatures T 1 and T 2 . A thermal stratification exists in the case of regular wallboards (a difference of 1.3 • C for the temperature maximums between the two probes).

This thermal stratification does not exist for the PCM wallboards. This must be due to higher natural convection effects. This effect improve the thermal comfort (by avoiding thermal stratification) and ,as we know, has never been observed before.

Figure 9 shows the evolutions of mean temperatures for the interior surfaces of the three modified walls, and for the cases with and without PCM material.

The PCM walls temperature fluctuations are lower than for the ordinary walls;

the evolutions of the east and west walls interior temperatures being similar.

It is interesting to emphasize that the temperature evolution curves with and without PCM have a phase difference of about 40mn. The last remark is valid for all the three cases tested: the PCM material "adds" inertia to the wall.

Concerning the east and west walls temperature curves, the PCM material causes a decrease of the temperature amplitude of about 3.5 • C. For the north wall, this decrease of the amplitude is about 2.8 • C. This value is lower than for east and west walls because the solar radiations are more important on the north wall. For the mid-season case, the decrement factor is f = 0.78.

Experimental results for the mid-season case

The figure 11 shows the evolutions of mean temperatures for the interior sur-faces of the three modified walls, and for the cases with and without PCM material. Concerning the east and west walls temperature curves, the PCM material causes a decrease of the amplitude of about 2.6 • C. For the north wall, this decrease is about 1.3 • C. For the winter case, the decrement factor is f = 0.73.

Experimental results for the winter case

The figure 13 shows the evolutions of mean temperatures for the interior surfaces of the three modified walls, and for the cases with and without PCM material. Concerning the east and west walls temperature curves, the PCM material causes a decrease of the temperature amplitude of about 2.9 • C. For the north wall, this decrease is about 1.9 • C.

Discussion

Three different external climates are tested to evaluate the potential of the PCM wallboards. For the three cases, the PCM wallboards allow to reduce the air temperature of the room, compared with regular wallboards. The decrement factor of the air temperature amplitude is varying between 0.73 and 0.78.

The lower decrement factor, which corresponds to the most efficient PCM use, concerns the winter case with the lower air temperature.

On the whole, the PCM tested enable to maintain the room air temperature within the comfort zone by decreasing the maximum air temperature of the room to a maximum value of 4.2 • C.This enhancement of the thermal comfort is more important if considering that the wall surface temperatures are also lower for the PCM wallboards than for the regular ones.

The last interesting observation concerns the thermal stratification of the room which exists for regular wallboards but not for the PCM wallboards. Of course, no thermal stratification means that natural convection exists in the room with PCM wallboards or is enhanced with PCM wallboards. Further investigations are needed to evaluate this effect, but it is not the purpose of our paper.

Conclusions

In order to investigate light building envelop, an experimental research has been carried out with wall containing PCM material. This is one of the rare study allowing a differential analysis of walls with and without PCM material, under controlled thermal and radiative effects. The tests concern the behavior of the test cell for summer, mid-season and winter cases.

Including the PCM wallboard reduces the air temperature fluctuations in the room. The decrement factor observed for the cases with PCM wallboard and with regular wall is about 0.7 for all the season tested. The wall surface temperatures fluctuations are also reduced.

For a building application, the PCM composite is interesting to enhance the human thermal comfort for three main reasons:

• The PCM material included in the walls strongly reduces the overheating effect (and the energy stored is released to the air room when the temperature is minimum).

• The wall surface temperature is lower when using PCM wallboard, then the thermal comfort is enhanced by radiative heat transfer.

• The natural convection mixing of the air is also enhanced by PCM material, avoiding uncomfortable thermal stratifications.

In order to validate the use of PCM wallboard for light weight building envelop, further investigations are needed. Numerical simulations must be carried out too, in order to investigate the total gain for real buildings. We think that the results described in this article can then be used as reference test-case for validations of numerical codes. We also need to look at modifications of the PCM wallboard position to enhance storage effects.

Finally, the experimental set-up is fully described sot that the data presented in this study can be used for the validation of PCM numerical modeling.

0.05Kmn - 1 .

 1 The thermal analysis is presented for the range [-20 • C; 35 • C]. Two curves are presented: the freezing curve (cooling from 35 • C to -20 • C) and the melting curve (heating from -20 • C to 35 • C).

3. 2

 2 figure 3 represents a scheme of the experimental set-up. The test room is composed of two identical enclosures called test cell 1 and 2. In our experiments, only the cell 1 has been used and then is called test cell in the rest of the paper.

  ), each line being tilted of an angle α: for line A α = 0 • , for the line B α = 25 • and for the line C α = 50 • . The radiative flux thus created penetrates in the cell via the glazed wall. The control makes it possible to dynamically control the level of radiative flux by the means of the number of lit spotlights.

4 . 1 Figure 8

 418 Figure 8 shows the room air temperature profiles for the PCM and regular wallboards. With regular wallboards, the air temperature is varying between a minimum of 18.9 • C and a maximum of 35.3 • C and 36.6 • C for the probes at respectively heights 85cm and 170cm. The difference between the two maximum temperature for the two probes at different heights is due to the thermal stratification of the room air. With PCM wallboards, the air temperature is varying between a minimum of 19.8 • C and a maximum of 32.7 • C. Then, the PCM included in the walls allows to reduce the temperature fluctuations in the room: the maximum air temperature value decreases of about 3.9 • C while the minimum air temperature increases of about 0.8 • C.

Figure 10 shows

 10 Figure 10 shows the room air temperature profiles for the PCM and regular wallboards. With regular wallboards, the air temperature is varying between a minimum of 17.4 • C and a maximum of 30.8 • C and 33.5 • C for the probes at respectively heights 85cm and 170cm. With PCM wallboards, the air temperature is varying between a minimum of 17.8 • C and a maximum of 30.7 • C and 29.0 • C for the probes at respectively heights 85cm and 170cm. Then, the PCM included in the walls allows to reduce the temperature fluctuations in the room: the maximum air temperature value decreases of about 2.3 • C while the minimum air temperature increases of about 0.4 • C.

Figure 12

 12 Figure 12 shows the room air temperature profiles for the PCM and regular wallboards. With regular wallboards, the air temperature is varying between a minimum of 18.6 • C and 18.1 • C and a maximum of 30.4 • C and 32.2 • C for the probes at respectively heights 85cm and 170cm. With PCM wallboards, the air temperature is varying between a minimum of 18.6 • C and 18.1 • C and a maximum of 28.7 • C and 28.4 • C for the probes at respectively heights 85cm and 170cm. Then, the PCM included in the walls allows to reduce only the maximum air temperature in the room of about 4.2 • C.

5 N 1 Fig. 3 .Fig. 4 .

 5134 Fig. 3. Scheme of the experimental set-up.

Fig. 5 .Fig. 7 .Fig. 8 .

 578 Fig. 5. Isometric diagram of the test cell with the 12 spotlights -Locations of the modified walls.

Fig. 9 .Fig. 10 .

 910 Fig. 9. Experimental mean temperatures of the interior surfaces of east wall (a), west wall (b) and north wall (c) -summer case with ventilation.

Fig. 11 .Fig. 12 .

 1112 Fig. 11. Experimental mean temperatures of the interior surfaces of east wall (a), west wall (b) and north wall (c) -mid-season case.

Fig. 13 .

 13 Fig. 13. Experimental mean temperatures of the interior surfaces of east wall (a), west wall (b) and north wall (c) -winter case.

Acknowledgement

The authors wish to thank Dupont de Nemours Society for their support and particularly the head of this project, Raymond Reisdorf.

Wall without PCM 1 2 34 Wall with PCM


G Chanfray

Magda Ericson

In this talk we present a description of nuclear binding in a chiral approach based on the existence of a chiral invariant scalar field associated with the generation of the masses through spontaneous chiral symmetry breaking. We discuss the emergence of such a field on the example of the NJL model. We also incorporate the effect of confinement at the level of the nucleon substructure to stabilize nuclear matter. In a particular quark-diquark model we illustrate the simutaneous influences of spontaneous chiral symmetry breaking and confinement on the nucleon mass and on the nuclear matter description.

Introduction. The relation between the fundamental properties of low energy QCD, namely chiral symmetry and confinement, and the nuclear many-body problem is certainly one of the most challenging aspect of present day nuclear physics. One particular crucial question is how the interplay between chiral symmetry and confinement in the nucleon structure manifests itself in the nuclear many-body problem. In this talk we will develop a viewpoint where the nuclear attraction is associated with the in-medium fluctuation of a (chiral invariant) background scalar field which is itself at the origin of the constituent quark mass. Nuclear stability is ensured with the incorporation of the nucleon response to this scalar field. This response depends on the quark confinement mechanism inside the nucleon. As we will see this question can be rephrased in terms of the relative weight of spontaneous chiral symmetry breaking and confinement in the origin of the nucleon mass.

Chiral nuclear matter description including the effect of confinement. As a starting point let us consider the relativistic mean-field approaches initiated by Walecka where the nucleons move in an attractive scalar and a repulsive vector background fields. It provides an economical saturation mechanism and a well known success is the correct magnitude of the spin-orbit potential where the large vector and scalar fields contribute in an additive way. Now the question of the very nature of these background fields and their relationship with the QCD condensates has to be elucidated. To address this question we formulate an effective theory parametrized in terms of the fields associated with the fluctuations of the chiral condensate in a matrix form (M = σ + i τ • π) by going from cartesian to polar coordinates, i.e., going from a linear to a non linear representation : [1] we made the physical assumption to identify the chiral invariant scalar field, s = S -f π , associated with radial (in order to respect chiral constraints) fluctuations of the condensate, with the background attractive scalar field. In this picture the nuclear medium can be seen as a shifted vacuum characterized by a chiral order parameter S = f π + s which governs the evolution of the nucleon mass (∆M N /M N = s /f π ) and part of the evolution of the chiral condensate. It turns out that this scalar field, which has derivatives coupling with the pion, largely decouples from low energy pions whose dynamics is governed by chiral perturbation theory. This scalar field is relevant for low momentum nuclear physics and the question of its relation with the famous σ(600) (which appears experimentally as a broad ππ resonance) is to a large extent irrelevant for our purpose.

M = σ + i τ • π = S U = (f π + s) exp i τ • Φ π /f π . In ref.
In our approach the Hartree energy density of nuclear matter (including omega exchange) writes in terms of the order parameter s = s :

E 0 /V = ε 0 = (4 d 3 p/(2π) 3 ) Θ(p F -p) E * p (s) + W (s) + g 2 ω /2m 2 ω ρ 2 , where E * p (s) = p 2 + M * 2 N (s)
is the energy of a nucleon with the effective Dirac mass, M * N (s) = M N + g S s and g S = M N /f π is the scalar coupling constant of the sigma model. Here two serious problems appear. The first one is the fact that the chiral effective potential, W (s), contains an attractive tadpole diagram which generates an attractive three-body force destroying matter stability [2]. The second one is related to the nucleon substructure. According to the lattice data analysis of Thomas et al [3], the nucleon mass can be expanded according to

M N (m 2 π ) = a 0 + a 2 m 2 π + a 4 m 4 π + Σ π (m π , Λ) + ...,
where the pionic self-energy is explicitely separated out. While the a 2 parameter is related to the non pionic piece of the πN sigma term, a 4 is related to the nucleon QCD scalar susceptibility. The important point is that (a 4 ) latt ≃ -0.5 GeV -3 [3] is essentially compatible with zero in the sense that it is much smaller than in our chiral effective model, (a 4 ) Chiral = -3f π g S /2 m 4 σ ≃ -3.5 GeV -3 , where the nucleon is seen as a juxtaposition of three constituent quarks getting their mass from the chiral condensate [4]. The common origin of these two failures can be attributed to the absence of confinement [5]. In reality the composite nucleon responds to the nuclear environment, i.e., by readjusting its confined quark structure [6]. The resulting polarization of the nucleon is accounted for by the phenomenological introduction of the positive scalar nucleon response, κ N S , in the nucleon mass evolution :

M N (s) = M N + g S s + 1
2 κ N S s 2 + ..... This constitutes the only change in the expression of the energy density but this has numerous consequences. In particular the a 4 parameter is modified :

a 4 = (a 4 ) Chiral 1 -2 3 C . The value of C ≡ (f 2 π /2 M N )κ N S which reproduces the lattice data is C ≃ 1.
25 implying a strong cancellation effect in a 4 [4,7]. Moreover the scalar response of the nucleon induces an new piece in the lagrangian L s 2 N N = -κ N S s 2 N N/2 which generates a repulsive three-body force able to restore saturation [8].

The restoration of saturation properties has been confirmed at the Hartree level [4] with a value of the dimensionless scalar response parameter, C, close to the value estimated from the lattice data, taking g S = M N /f π and g ω adjusted near the VDM value. The next step has been to include pion loops on top of the Hartree mean-field calculation [7,9]. One possibility is to use in-medium chiral perturbation theory but we prefer to use a standard many-body (RPA) approach which includes the effect of short-range correlations (g ′ parameters fixed by spin-isospin phenomenology) and ∆ -h excitations. The calculation which also incorporates rho exchange has no real free parameters apart for a fine tuning of C (around the lattice estimate) and of g ω (around the VDM value). We have The inclusion of spin-isospin loops improves the quality of the result, in particular they bring down the compressibility close to the accepted value. We stress the relatively modest value of the correlation energy (≃ -10 MeV ), much smaller than what is obtained from iterated pion exchange (planar diagramm) in in-medium chiral perturbation theory. This effect is mainly due to the strong screening of pion exchange by short-range correlations.

We have also performed a full relativistic Hartree-Fock calculation with the notable inclusion of the rho meson exchange which is important to also reproduce the asymetry properties of nuclear matter [5]. Again in an almost parameter free calculation, saturation properties of nuclear matter can be reproduced with g S = 10, m σ = 800 MeV , g ρ = 2.6 (VDM), g ω = 6.4 (close to the VDM value 3 g ρ ) and C = 1.33 (close to the lattice value). An important conclusion is that the rho Hartree contribution (7 MeV ) to the asymmetry energy a S is not sufficient to reproduce the accepted value of a S around 30 MeV when keeping the VDM value for the vector coupling constant, g ρ . The Fock term through its tensor contribution is absolutely necessary and the agreement is even better when κ ρ is increased from the VDM value, κ ρ = 3.7 to κ ρ = 5 (strong rho scenario). The model also predicts a neutron mass larger than the proton mass with a difference increasing with neutron richness in agreement with ab-initio BHF calculations [10].

Status of the nuclear scalar field. Since the very notion of a scalar field has met some septicism it is important to remind how such an object naturally emerges. For this purpose we introduce the NJL model in the light quark sector limited to describe the main mesons : the pion, the sigma, the rho, the a 1 and the omega mesons. The lagrangian is :

L = ψ (i γ µ ∂ µ -m) ψ + (G 1 /2) ψψ 2 + ψ iγ 5 τ ψ 2 -(G 2 /2) ψ γ µ τ ψ 2 + ψ γ µ γ 5 τ ψ 2 + ψ γ µ ψ 2 .
It is generally accepted that this model gives an excellent description of vacuum chiral symmetry breaking, with the appearence of a constituent quark mass and a soft Goldstone pion mode. In addition in the so called delocalized version the inclusion of a form factor, depending on a cutoff, Λ, at each quark leg of the interaction generates a momentum dependence of the quark mass in agreement with lattice calculation. However the approach is notoriously not fully satisfactory due to the lack of confinement : in particular unphysical decay channels of vector mesons in q q pairs may appear. In that respect it has been shown that adding a confining interaction on top of the NJL model solves the problem of unphysical qq decay channels of mesons with masses larger than twice the constituent quark mass [11]. In addition the scalar meson arising around twice the constituent quark mass is pushed at higher energy by confinement and the σ(600) comes out as a broad ππ resonance. It has also been demonstrated that the effect of the confining interaction, although crucial for the on-shell properties of vector and scalar mesons, has only little influence for the low momentum physics relevant for the nuclear many-body problem [12]. Hence, our attitude will be to derive an effective low momentum theory where the mass parameters for scalar et vector mesons will not be the on-shell mesons masses but simply mass parameters associated with the inverse of the corresponding correlators taken at zero momentum. Technically this can be done by rewriting the NJL lagrangian in a semi-bozonized form and integrating out quarks in the Dirac sea using a path integral formalism. The physical meaning is simply a projection of q q vacuum fluctuations onto meson degrees of freedom. Keeping only the relevant terms for nuclear physics purpose, the resulting low momentum effective lagrangian has the form :

L = 1 2 I l 2S ( S) I l 2S (M 0 ) ∂ µ S∂µS -W (S = g 0S S) + 1 4 F 2 M 2 π S M 0 tr f (U + U † -2) + 1 2F 2 Ĩ( S) S2 ∂ µ Φ∂ µ Φ + 1 2 M 2 V (ω µ ω µ + v µ • v µ ) + 1 2 M 2 A ( a µ • a µ ) - 1 4 (ω µν ω µν + v µν • v µν + a µν • a µν ) (1)
S is a chiral invariant scalar field whose vacuum expectation value coincides with the constituent quark mass M 0 and S is the canonical one. We also introduce an effective scalar field (S) ef f = (F π /M 0 )S normalized to F π in the vacuum as in our previous approaches and W is the corresponding chiral effective potential. The matrix U is U = exp(Φ/F ) where Φ is the orthoradial pion field and the parameter F can be identified with the pion decay constant parameter F π . The quantity S = (M 0 /F π )(F π + s) is the expectation value of the scalar field which is expected to decrease in the nuclear medium. The various I's functions are standard NJL loop integrals which also enter the mass parameters M V and M A for vector and axial-vector mesons. These masses also depend on the vector coupling constant G 2 . In particular the ratio g 2 V /M 2 V (g V being the quark-vector coupling constant) is fixed to G 2 . Finally the pion mass emerges as

M 2 π = m M 0 /G 1 F 2 π .
We have a priori four parameters, G 1 , G 2 , the cutoff Λ and the bare quark mass m. We use : Λ = 1 GeV, m = 3.5 MeV, G 1 = 7.8 GeV -2 . We thus obtain for the vacuum quark mass at zero momentum : M 0 = 371 MeV and qq = -(286 MeV ) 3 . We actually constrain G 2 to be close to the VDM value (G 2 ) V DM = g 2 V /M 2 V = (2.65/0.770) 2 GeV -2 . For the nuclear matter calculation we take G 2 = 0.78 (G 2 ) V DM ⇒ F π = 93.6MeV, M π = 137.8 MeV . With this set of parameters the low momentum mass parameters are : (M σ ) ef f = 659 MeV , M V = 653 MeV and M A = 797 MeV .

Effect of confinement : a toy model for the nucleon. We now come to the second imortant point of this paper, namely the scalar response of the nucleon. Clearly a nucleon made of the simple adjonction of three NJL constituent quarks will not exhibit a scalar response since the confinement mechanism is lacking. The question of the scalar response is intimately related to the respective role of chiral symmetry breaking and confinement in the generation of the nucleon mass. One possibility is to reduce the relative role of chiral symmetry breaking by considering a nucleon made of a quark and a sufficiently light diquark to leave room for the confinement. As discussed in a set of works of Bentz et al (see ref. [2]), it is possible to construct from the NJL model a nucleon which has a diquark component. The diquark mass is also medium dependent since it depends on the constituent quark mass. If we take for the the coupling constant in the diquark channel, G1 = 0.92 G 1 , we obtain for the vacuum diquark M D = 398.5 MeV which turns out to be nearly equal to the constituent quark mass. In [2], a nucleon scalar response is obtained through the inclusion of an infrared cutoff µ R ≃ 200MeV in the Schwinger proper time regularization scheme. Such a prescription implies that quarks cannot propagate at relative distance larger than 1/µ R , hence mimicking a confinement mechanism. Here we propose to incorporate confinement in a more direct way. Since the diquark is in an anti-triplet color state, it is physically plausible that a string develops between the quark and the diquark as in a Q Q meson. We thus introduce a confining potential between the quark and the diquark: V (r) = K r 2 . In the non relativistic limit, the problem reduces to solve the Schrodinger equation for a particle with reduced mass µ, placed in an harmonic potential. In this limit the mass of the (in-medium) nucleon is given by :

M N ( S) = M( S) + M D ( S) + 3 2 K µ( S) with µ = M M D M + M D
Taking for the string tension a standard value K = (290 MeV ) 3 , we obtain for the vacuum nucleon mass M N = 1304 MeV . Thus the nucleon mass receives about equal components from chiral symmetry breaking and confinement. The scalar coupling constant of the nucleon to the effective scalar field (normalized to

F π in the vacuum) is (g S ) ef f ( S) = (M 0 /F π )(∂M N /∂ S) S=M 0 .
Its vacuum value is (g S ) ef f = 7.14. What really matters for the magnitude of the scalar attractive potential is the ratio (g S ) ef f /(M σ ) ef f = 7.15/659MeV which is close to the value inspired from the linear sigma model used in or previous works, 10/850MeV . The other important point is that (g S ) ef f decreases when the scalar field deviates from its vacuum value, i.e., when s = S -F π increases in absolute value at finite density. This translates into the fact that the nucleon masses stabilizes and even starts to increase with increasing nuclear scalar field s (see fig. 1). The pion nucleon sigma term can be obtained using the Feynman-Hellman theorem. Its numerical value is σ (nopion) σ = 30 MeV . Of course it also receives another contribution from the pion cloud. According to our previous works [7] we expect: σ (pioncloud) ≃ 20 MeV which corresponds to a pion cloud self-energy of -420 MeV . This is the order of magnitude needed to decrease the nucleon mass from 1.3 GeV to the physical value. In the following we will substract the nucleon mass by 1304 -940 = 364 MeV which we attribute to the pion cloud. In fig. 1 we show the result of the calculation of the various masses versus s/F π .

We are now in position to calculate the energy of symmetric nuclear matter in the Hartree approximation. The energy density writes :

E 0 V = ε 0 = 4 d 3 p (2π) 3 Θ(p F -p) p 2 + M 2 N ( S) -(M N ) vac + W ( S) + 9 G 2 2 ρ 2 . ( 2
)
The expectation value for the scalar field is obtained by minimization of the energy density, ∂ε 0 /∂ S = 0. The resulting curve (fig. 1) displays a saturation mechanism driven by the scalar nucleon response but the binding is not sufficient unless we decrease artificially the vector coupling constant G 2 at a value much smaller than the VDM result. We add on top of the Hartree mean field result the pion loop (Fock term and correlation energy) contribution obtained in our previous work [7]. Taking G 2 = 0.78 (G 2 ) V DM , we obtain a very decent saturation curve shown in fig. 2. Of course a fully consistent calculation within the model of the pion loop energy would certainly modify the result but a fine tuning on G 2 would presumably be sufficient to recover the correct saturation curve. The lesson of this simple model calculation seems to confirm our previous conclusions. The confinement effect (scalar response of the nucleon) is able to stabilize nuclear matter and the pion loop correlation energy helps to get the correct binding energy. We finally stess that the whole approach relies on the existence of a chiral invariant scaler field, not reducible to an effective two-pion state, but intimately related to the generation of the masses through spontaneous chiral symmetry breaking.

  Pacs: 24.85.+p 11.30.Rd 12.40.Yx 13.75.Cs 21.30.-x

Figure 1 :

 1 Figure 1: Left panel: Mass of the quark (dashed line), of the diquark (dot-dashed line) and of the nucleon (full line) versus the relative deviation of the scalar field with respect to its vacuum value. Right panel: Binding energy of nuclear matter versus nuclear matter density in units of normal density. The full line corresponds to the full result and the dashed line represents the Hartree result. The dot-dashed line corresponds to the contribution of the Fock term and the dotted line represents the correlation energy. All the numerical inputs are given in the text.

Acknowledgments. It is a pleasure to mention that many ideas developped in this talk have been influenced by the works of Tony Thomas on the lattice analysis of the nucleon mass and on the implications of the QCD nucleon structure on the nuclear many-body problem.


Pierre-Vincent Koseleff
email: koseleff@math.jussieu.fr

Daniel Pecker
email: pecker@math.jussieu.fr

P. -V Koseleff

Chebyshev knots

Keywords: Polynomial curves, Chebyshev polynomials, Chebyshev curves, Lissajous knots, long knots, braids Mathematics Subject Classification 2000: 14H50, 57M25, 14P99

come

Introduction

A Lissajous knot is a knot which admits a one-to-one parametrization of the form x = cos(at); y = cos(bt + ϕ); z = cos(ct + ψ) where 0 ≤ t ≤ 2π and where a, b, c are pairwise coprime integers. These knots, first defined in [START_REF] Bogle | Lissajous knots[END_REF], have been studied by many authors: V. F. R. Jones, J. Przytycki, C. Lamm, J. Hoste and L. Zirbel. Most known properties of Lissajous knots are deduced from their symmetries, which are easy to see (see [START_REF] Jones | Lissajous knots and billiard knots[END_REF][START_REF] Lamm | There are infinitely many Lissajous knots[END_REF][START_REF] Hoste | Lissajous knots and knots with Lissajous projections[END_REF][START_REF] Przytycki | Symmetric knots and billiard knots, Chapter 20 of the book Ideal Knots[END_REF]).

On the other hand Vassiliev considered polynomial knots, i.e. non singular polynomial embeddings R → R 3 (see [START_REF] Vassiliev | Cohomology of knot spaces, Theory of Singularities and its Applications[END_REF][START_REF] Shastri | Polynomial representation of knots[END_REF][START_REF] Ranjan | On polynomial representation of torus knots[END_REF][START_REF] Koseleff | On polynomial torus knots[END_REF][START_REF] Koseleff | A polynomial parametrization of torus knots[END_REF]).

In this paper we study a polynomial analogue of Lissajous knots. It is natural to use the classical Chebyshev polynomials T n (t) instead of cosine functions to define our Chebyshev knots. The Chebyshev polynomials are defined by T 0 = 1, T 1 = t, T n+1 = 2tT n -T n-1 , n ∈ Z. They satisfy the trigonometric identity cos(n θ) = T n (cos θ).

Definition 1 A knot in R 3 ⊂ S 3 is a Chebyshev knot if it admits a one to one parametrization of the form x = T a (t); y = T b (t); z = T c (t + ϕ)

where t ∈ R, a, b, c are integers and ϕ is a real constant.

Figure 1: The mirror image of the 7 7 knot is Chebyshev.

We begin with the study of plane Chebyshev curves which are projections of Chebyshev knots on the (x, y)-plane. We conclude this paragraph with a theorem of Hoste and Zirbel [START_REF] Hoste | Lissajous knots and knots with Lissajous projections[END_REF] describing these curves in terms of particular braid projections.

Then, we study some families of Chebyshev knots with ϕ = 0, called harmonic knots. We prove that for a, b coprime positive integers and c = abab, the harmonic knot H(a, b, c) has an alternating projection on the (x, y)-plane. We deduce that there are infinitely many harmonic knot types. This is similar to a theorem of C. Lamm concerning Lissajous knots (see [START_REF] Lamm | There are infinitely many Lissajous knots[END_REF]). We also prove that the torus knots T(2, 2n + 1) are harmonic knots. On the other hand, we observe that the symmetries of harmonic knots are quite different from those of Lissajous knots. There are infinitely many amphicheiral harmonic knots and infinitely many strongly reversible harmonic knots. The trefoil and the figure-eight knot are harmonic knots but are not Lissajous. Some knots are both Lissajous and harmonic knots, e.g. 5 2 and 7 5 .

We conclude the paper with our principal result: every knot is a Chebyshev knot. This is done by showing first that every knot has a plane projection which is a Chebyshev curve. Then we use some classical results of braid theory and a density argument based on Kronecker's theorem.

At the end we give Chebyshev diagrams of the first 2-bridge harmonic knots.

Geometry of plane Chebyshev curves

Chebyshev curves were defined in [START_REF] Fischer | Plane Algebraic Curves[END_REF] to replace the older denomination of "doubly parametrized Lissajous curves". Their double points are easier to study than those of Lissajous curves. It will be convenient to consider also the case of implicit Chebyshev curves.

= (-1, 1) 2 , R = {(x, y) ∈ Q, T b (x) = T a (y) = 1}, and R ′ = {(x, y) ∈ Q, T b (x) = T a (y) = -1}.
Proof. The singular points of C are obtained for T ′ b (x) = 0, T ′ a (y) = 0, T b (x) = T a (y). From T a (cos θ) = cos aθ, we deduce that T a has degree a and T ′ a (cos θ) = a sin aθ sin θ . T ′ a has a -1 simple roots in (0, 1):

y k = cos k π a , k = 1, . . . , a -1.
At these points, we have

T a (y k ) = (-1) k . T ′ b has b -1 roots in (0, 1): x 1 , . . . , x b-1 .
For each x i there are exactly

1 2 (a -1) values y j satisfying T ′ a (y j ) = 0, T a (y j ) = T b (x i ).
Hence the number of singular points is 1 2 (a -1)(b -1), and they form two rectangular grids. Since the roots of T ′ b (x) = are simple, we see that these points are crossing points.

Remark 1 It follows from their definitions that

|R| = 1 2 b-1 2 (a -1), |R ′ | = 1 2 b 2 (a -1)
where ⌊x⌋ is the greatest integer less than or equal to x.

Proposition 2 Let a and b are nonnegative coprime integers, a being odd. Let the Chebyshev curve C be defined by the equation T b (x)-T a (y) = 0. Then C admits the parametrization x = T a (t), y = T b (t). The pairs (t, s) giving a crossing point are

t = cos k a + h b π, s = cos k a - h b π,
where k, h are positive integers such that

k a + h b < 1. Proof. Since T a • T b = T b • T a = T ab , the rational curve C ′ parametrized by x = T a (t), y = T b (t) is contained in C. These two curves intersect the line {x = x 0 } in one point if |x 0 | > 1, in a points if |x 0 | < 1 and in 1 2 (a + 1) points if x 0 = ±1. Consequently, they are equal. The 1 2 (a -1)(b -1) pairs t = cos k a + h b π, s = cos k a - h b π,
give rise to double points of C ′ = C. Because the number of singular points of C is 1 2 (a -1)(b -1), we see that there is no other singular point.

Remark 2 We observe that the crossing points are obtained for the (a -1)(b -1) elements of

E = {t u = cos u ab π, 0 ≤ u ≤ ab, a | u, b | u}. ( 1
)
For these values, we get T b (x(t u )) = T a (y(t u )) = (-1) u . Note that t u and t u ′ correspond to the same point when u ≡ -u ′ (mod 2b) and u ≡ u ′ (mod 2a).

Remark 3 In general, the curve C : T b (x) -T a (y) = 0 has d 2 + 1 components where d = gcd(a, b). See Figure 4.

The following proposition will be useful to consider Chebyshev curves as trajectories in a rectangular billiard (see [START_REF] Jones | Lissajous knots and billiard knots[END_REF]).

Proposition 3 Let C be the Chebyshev curve: T b (x) -T a (y) = 0. There exists an homeomorphism from the square

I 2 = [-1, 1] 2 to the rectangle [0, b] × [0, a],
such that the image of C I 2 is the union of all the billiard trajectories with slopes ±1 through the points with coordinates x = b, y = a -2k, 0 ≤ 2k ≤ a.

Proof. Consider the mapping F (x, y) = (X, Y ) with πX = b(πarccos x), πY = a(πarccos y). By trigonometry, it is not hard to check that F has the announced properties. We shall now present a description of Chebyshev curves using braids. Let B n be the group of braids on n strings. For practical purposes we shall draw these braids horizontally, the strings being numbered from the bottom to the top. The standard braid generators are denoted σ 1 , σ 2 , . . . , σ n-1 . The braid σ i exchanges the strings i + 1 and i, the string i + 1 passing over the string i. In this paragraph we shall be interested in plane projections of braids, called plane braids. We shall also consider the composition of such plane braids. Let a = n. Let s i denote the plane braid which is the plane projection of σ i . This plane braid has one crossing point. Following Hoste and Zirbel [START_REF] Hoste | Lissajous knots and knots with Lissajous projections[END_REF], let us define the plane braids s even and s odd as

s even = s 2 s 4 • • • s E , s odd = s 1 s 3 • • • s O ,
where E and O are the largest even and odd integers less than n = a.

R ε = {|x| < 1 -ǫ, |y| ≤ 1}. Then there is a homeomorphism between the pairs (R ε , C) and (R ε , ρ) where ρ = (s odd s even ) b-1 2 if b is odd and ρ = (s even s odd ) b-2 2 s even if b is even. Proof. Following the proof of Proposition 1, the 1 2 (a -1)(b -1) singular points of C are in R ε when ε is small enough. For each k = 1, . . . , b -1, there are 1 2 (a -1) singular points (x k , y l ) = cos k π b , cos l π a , k + l ≡ 0 (mod 2).
It means that over a neighborhood over x k , the curve C is isotopic to s even if k is odd, and isotopic to s odd if k is even. This proves the result. 2

We can define the plat closure of a plane horizontal braid with 2m strings labelled 0, 1, . . . , 2m-1, to be the plane curve obtained by connecting the right ends 0 to 1, . . . , 2m -2 to 2m -1, and the left ends in the same order.

Corollary 1 Let a be an odd integer, and b an even integer. Let ρ ′ be the plane braid with a + 1 strings obtained by adding a free string numbered a + 1 over ρ = (s even s odd )

b-2

2 s even . Then the Chebyshev curve T b (x) -T a (y) = 0 is isotopic (in S 2 ) to the plat closure of the plane braid ρ ′ .
Proof. Let us illustrate this by looking at the curve T 10 (x) -T 5 (y) = 0, which has 3 components. We see on Figure 4 that it is the plat closure of (s even s odd ) 4 s even .

2

Harmonic knots

In this paragraph we shall study Chebyshev knots with ϕ = 0. Comstock (1897) found the number of crossing points of the harmonic curve parametrized by x = T a (t), y = T b (t), z = T c (t). In particular, he proved that this curve is non singular if and only if a, b, c are pairwise coprime integers [START_REF] Comstock | The Real Singularities of Harmonic Curves of three Frequencies[END_REF]. Such curves will be named harmonic knots H(a, b, c).

We see that H(a, b, 1) is the unknot because the height function is monotonic. We can also obtain the unknot in a less trivial way. By trigonometry, we see that the bounded part of our knot is on the surface

S = {(x, y, z) ∈ R 3 , |x| ≤ 1, |y| ≤ 1, z = xy ± (1 -x 2 )(1 -y 2 )}.
Since S is the union of two sheets that are homeomorphic to the square [0, 1] 2 glued along their boundaries, we see that it is homeomorphic to a sphere. Consequently the genus of H(a, b, c) is zero, hence it is the unknot. 2

Note that the surface S has the symmetries of a regular tetrahedron. It is contained in the cubic surface {x 2 + y 2 + z 2 = 1 + 2xyz} which has the same symmetries.

Let C be a plane projection of a parametrized knot. Consider a crossing point of C obtained for the parameter pair (t, s). The tangents at this point have opposite slopes (see [START_REF] Koseleff | The first rational Chebyshev knots[END_REF], Lemma 4). It follows easily that the nature of this crossing point depends only of the sign of the expression D = z(t)z(s) x ′ (t)y ′ (t). This is not the usual definition of the sign of oriented crossings, see Figure 6. Proof. The crossing points of the plane projection {x = T a (t), y = T b (t)} are obtained for the parameters t = cos τ, s = cos σ, where

D > 0 D < 0
τ = k a + h b π, σ = k a - h b π.
Using trigonometry we get x ′ (t) = a sin aτ sin τ , y ′ (t) = b sin bτ sin τ , so

x ′ (t)y ′ (t) = (-1) h+k ab sin 2 τ sin( ah b π) sin( bk a π). (2)
We have also

z(t) -z(s) = T c (t) -T c (s) = -2 sin ch b π sin ck a π . (3)
and the announced result. 2

Alternate harmonic knots

The following theorem is the analogue of a theorem of Lamm [START_REF] Lamm | There are infinitely many Lissajous knots[END_REF] concerning Lissajous knots. The polynomial x ′ (t)y ′ (t) has (a + b -2) simple roots for t u = cos u ab π, where a or b divides u = 1, . . . , ab -1. For these parameters, the billiard curve corresponding to the (x, y)-plane projection bounces on a wall.

Three cases may occur because at least one of three consecutive t u belongs to E.

1. t h+1 ∈ E and t h ∈ E. Then sign x ′ (t h )y ′ (t h ) = sign x ′ (t h+1 )y ′ (t h+1
) and since the sign of D changes, we conclude that the sign of z(t)z(s) changes between the 2 consecutive parameters t h+1 and t h .

2. t h+1 ∈ E, t h ∈ E, t h-1 ∈ E. We have x ′ (t)y ′ (t) = 0 at t h and t h-1 . For t h-1 < t < t h-2 , we have sign x ′ (t)y ′ (t) = sign x ′ (t h+1 )y ′ (t h+1 ) , so sign x ′ (t h-2 )y ′ (t h-2 ) = sign x ′ (t h+1 )y ′ (t h+1
) . Hence we see that the sign of z(t)z(s) changes between the 2 consecutive parameters t h+1 and t h-2 .

3.

t h+1 ∈ E, t h ∈ E, t h-1 ∈ E. We have x ′ (t)y ′ (t) = 0 at t h , so sign x ′ (t h-1 )y ′ (t h-1 ) = -sign x ′ (t h+1 )y ′ (t h+1
) . Hence we see that the sign of z(t)z(s) changes between the 2 consecutive parameters t h+1 and t h-1 .

In conclusion, the diagram is alternating. 2

Symmetries and harmonic knots

A knot K in S 3 is strongly (-)amphicheiral if there is an involution of (S 3 , K) which reverses the orientation of both S 3 and K. A knot K in S 3 is strongly reversible (or strongly invertible) if there is an involution of (S 3 , K) which preserves the orientation of S 3 and reverses the orientation of K (see [START_REF] Kawauchi | A Survey of Knot Theory[END_REF], pp. 127-128).

Proposition 6

The harmonic knot H(a, b, c) is either strongly (-)amphicheiral if abc is odd, or strongly reversible if abc is even.

Proof. It is immediate from the parity of Chebyshev polynomials. 2

Corollary 2 There are infinitely many amphicheiral harmonic knots. There are infinitely many strongly reversible harmonic knots.

Proof. Since the harmonic knot H(a, b, c), c = abab is alternate, its crossing number is 1 2 (a -1)(b -1). From this we conclude that there is an infinity of such knots with a, b, c odd, or with abc even.

2

If σ is any permutation of {a, b, c} then the harmonic knot H(σ(a), σ(b), σ(c)) is either H(a, b, c) if σ is an even permutation or its mirror image if σ is an odd permutation. Proof. The expression of sign D (Equations ( 2) and ( 3)) for a given pair of parameters (t, s) corresponding to crossing points in the (x, y)-plane projection is invariant under the transformation c → c + 2ab and c → 2abc. 2

We can therefore suppose that a < b and 0 < c < ab to consider all cases.

Independently but later, G. and J. Freudenburg ([6]) proved the following improvement of

The simplest alternate harmonic knots

It is remarkable that for a = 3 the curves are drawn in "Conway normal form" for 2-bridge knots [START_REF] Murasugi | Knot Theory and its Applications[END_REF]. Then their Conway notation is H(3, n, 2n -3) = C(1, 1, . . . , 1) when n is not a multiple of 3. Turner [START_REF] Turner | On a class of knots with Fibonacci invariant numbers[END_REF] named these knots Fibonacci knots, because their determinants are Fibonacci numbers. For n = 4 we obtain the trefoil, for n = 5 the figure-eight, for n = 7 the 6 3 knot, and for n = 8 the 7 7 knot.

The Fibonacci knots with an even crossing number are 2-bridge amphicheiral knots. We have recently proved ( [START_REF] Koseleff | On Fibonacci knots[END_REF]) that they are not Lissajous.

For a = 4, we also obtain 2-bridge knots. Following the classical method ( [START_REF] Murasugi | Knot Theory and its Applications[END_REF] p. 183-187), we see that their Conway notation is H(4, n, 3n -4) = C(-1, -2, . . . , -1, -2) = C(-3, -1, -2, . . . , -1, -2), (n odd). For n = 5 we obtain the 6 2 knot, for n = 7 a symmetric picture of the 9 20 knot (compare with Rolfsen's table [START_REF] Rolfsen | Knots and Links[END_REF]).

For a ≥ 5 we may obtain p-bridge knots, with p ≥ 3. For example the harmonic knot H [START_REF] Fischer | Plane Algebraic Curves[END_REF][START_REF] Freudenburg | Curves defined by Chebyshev polynomials[END_REF][START_REF] Pecker | Simple constructions of algebraic curves with nodes[END_REF] is the mirror image of 10 116 knot in Rolfsen's table (amazingly, with exactly the same picture). Its bridge number is known to be 3.

Note that the torus knot T(2, 2n + 1) cannot be obtained as an alternate harmonic knot, except for the trefoil knot, as it is proved in [START_REF] Koseleff | On polynomial torus knots[END_REF]. Nevertheless it can be obtained as an harmonic knot. For k = 0, . . . , n -1, let us consider

A k be obtained for m = 3k + 1, m ′ = 2b -m. B k be obtained for m = 2b + 3k + 2, m ′ = m -2b.
C k be obtained for m = 2b + 3k + 3, m ′ = 4bm.

Then we have

x(A k ) = cos( 3k + 1 b π), y(A k ) = 1 2 (-1) k . x(B k ) = cos( 3k + 2 b π), y(B k ) = 1 2 (-1) k+1 . A 0 B 0 C 0 A 1 A n-1 B n-1 C n-1 Figure 11: H(3, 3n + 1, c), n even x(C k ) = cos( 3k + 3 b π), y(C k ) = 1 2 (-1) k .
Hence our 3n points satisfy

x(A k-1 ) > x(B k-1 ) > x(C k-1 ) > x(A k ) > x(B k ) > x(C k ), k = 1, . . . , n -1.
Let us determine the nature of the crossing points. Using the identity T ′ a (cos τ ) = a sin aτ sin τ , we get sign x ′ (t)y ′ (t) = sign sin 3τ sin bτ .

We get

for A k : sign x ′ (t)y ′ (t) = sign sin( 3k + 1 b π) sin( 3k + 1 3 π) = (-1) k . for B k : sign x ′ (t)y ′ (t) = sign sin 2b + 3k + 2 b π sin 2b + 3k + 2 3 π = (-1) k+1 sign sin 3k + 2 b π sin π 3 = (-1) k+1 . for C k : sign x ′ (t)y ′ (t) = sign sin( 2b + 3k + 3 b π) sin( 2b + 3k + 3 3 π) = (-1) k+1 sign sin( 3k + 3 b π) sin 2π 3 = (-1) k+1 .
Now, let us compute the sign of

T c (t) -T c (s) = -2 sin(c τ + σ 2 ) sin(c τ -σ 2 ) = -2 sin c 6b (m + m ′ )π sin c 6b (m -m ′ )π We have, with c = b + 1 = 3n + 2, for A k : z(t) -z(s) = -2 sin c π 3 sin c m -b 3b π = -2(-1) n sin 2π 3 sin c n -k b π = 2(-1) n sin 2π 3 (-1) n-k sin n -k b π so sign z(t) -z(s) = (-1) k . for B k : z(t) -z(s) = -2 sin c b + 3k + 2 3b π sin(c π 3 ) = -2 sin (n + k + 1 + n + k + 1 b )π (-1) n sin 2π 3 = 2(-1) k sin n + k + 1 b π sin 2π 3 . so sign z(t) -z(s) = (-1) k . for C k : z(t) -z(s) = -2 sin c 2π 3 ) sin(c k + 1 b π = -2 sin 4π 3 sin (b + 1) k + 1 b π = -2 sin 4π 3 )(-1) k+1 sin k + 1 b π so sign z(t) -z(s) = (-1) k+1 .
Collecting these results we finally get

sign D(A k ) = 1, sign D(B k ) = -1, sign D(C k ) = 1.
The Conway sequence of signs is then s(C n-1 ), s(B n-1 ), s(A n-1 ), . . . , s(A 0 ) with s(C k ), s(B k ), s(A k ) = (-1) n-1-k . Consequently the Conway normal form of our knot is C(1, 1, 1, -1, -1, -1, . . . , (-1) n-1 , (-1) n-1 , (-1) n-1 ). Its Schubert fraction is 2n + 1 2n ∼ -(2n + 1) and our knot is the torus knot T(2, 2n + 1). 2

Remark: Note that H(3, 3n + 1, 3(3n + 1) -1) is the mirror image of H(3, 3n + 1, 3n + 2). See Proposition 8.

Remark:

In [START_REF] Koseleff | A polynomial parametrization of torus knots[END_REF], we obtained the torus knot T(2, 2n + 1) as an alternate polynomial knot where x(t) = T 3 (t), y(t) = P (t), z(t) = Q(t) are polynomials and deg

P = 3n + 1, deg Q = 3n + 2
, that is to say the same degrees.

Because of their definitions, the symmetries of the harmonic knots are easy to find. They are either strongly negative amphicheiral if a, b, c are odd, or strongly reversible. So that not every knot is an harmonic knot. We can also remark that harmonic knots are billiard knots in a convex (compact) billiard (in fact a truncated cube) [START_REF] Jones | Lissajous knots and billiard knots[END_REF]. On the other hand, it is not difficult to see that if we change the nature of one crossing point in the diagram of the 10 116 knot, we can obtain the 8 17 knot. The knot 8 17 is famous because it is the first non reversible knot.

In the next paragraph, we shall see that it is possible to choose the nature of the crossing points with a (shifted) Chebyshev polynomial as height function.

Every knot is a Chebyshev knot

Let us denote B n the group of n-braids and S n the symmetric group. The group of pure braids P n is the kernel of the morphism π : B n → S n . If α is a braid, we shall denote ρ(α) its plane projection. In the next theorem, which is analogous to a theorem of Lamm for Lissajous curves (see [START_REF] Boocher | Sampling Lissajous and Fourier knots[END_REF][START_REF] Lamm | Cylinder knots and symmetric unions (Zylinder-knoten und symmetrische Vereinigungen)[END_REF]), we show that Theorem 3 Every knot has a projection which is a Chebyshev plane curve. This is a consequence of the following proposition.

Proposition 9 Let K be a knot, br(K) its bridge number. Let m ≥ br(K) be an integer. Then K has a projection which is a Chebyshev curve x = T a (t), y = T b (t), where a = 2m-1, and b ≡ 2 (mod 2a).

Proof. Let K be a knot. Let D be a regular diagram of K such that the abscissa has only two extremal values reached at m maxima and m minima. It means that K is the plat closure of a horizontal braid t with 2m strings. We can suppose the last string unbraided. Furthermore, reordering if necessary the ordinates of the 2m extrema, we can suppose that

π(t) = (2, 3) • • • (2m-2, 2m-1) = π(σ 2 σ 4 • • • σ 2m-2 ). Let us denote s even = ρ(σ 2 • • • σ 2m-2 ), and s odd = ρ(σ 1 • • • σ 2m-3 ).
As π(t) = π(σ 2 • • • σ 2m-2 ) we see that there exists l ∈ ker(π) = P 2m-1 , the group of pure braids, such that t = l • σ 2 • • • σ 2m-2 . As the braids A i j = x -1 σ 2 i x, where x = σ i+1 • • • σ j , generate P 2m-1 , the braid l is a composition of such elementary braids. It is not difficult to see that there is a braid α i j equivalent to A i j with plane projection ρ(α i j ) = (s even s odd ) a . Consequently, the braid t is equivalent to a braid projecting on (s even s odd ) ka s even . Using the braid description of Chebyshev curves (corollary 1), we conclude that our knot K is equivalent to a knot projecting upon the Chebyshev curve x = T a (t), y = T b (t), a odd, b ≡ 2 (mod 2a).

2

Figure 12: The "plane braid" (s even s odd ) 5 is the projection of the braid α 2,5 ∈ B 5 .

We shall prove our principal result with a density argument based on Kronecker's theorem ( [START_REF] Hardy | An Introduction to the Theory of Numbers[END_REF], Theorem 443, p. 382.) Let us recall this theorem.

Theorem 4 (Kronecker) If θ 1 , θ 2 , . . . , θ k , 1 are linearly independent over Q, then the set of points ((n θ 1 ), (n θ 2 ), . . . , (n θ k )) is dense in the unit cube. Here (x) denotes the fractional part of x.

We shall need the following lemma.

Lemma 2 Let c 1 , . . . , c k be real numbers such that -1 < c 1 < c 2 < • • • < c k < 1.
There exists a positive number e < 1c k such that the numbers arccos(c 1 + e), arccos(c 2 + e), . . . , arccos(c k + e), 1 are linearly independent over Q.

Proof. First, we shall prove, by induction on k, that the functions arccos(c i + x), 1 are linearly independent over R. Let λ 0 + k i=1 λ i arccos(c i +x) = 0 be a linear relation between these functions. We get by derivating

k i=1 λ i 1 1 -(c i + x) 2 = 0.
Then, when x → 1c k , we get λ k = 0, and the result follows by induction.

Suppose now that for each e < 1c k there exists a relation

λ 0 + k i=1 λ i arccos(c i + e) = 0, λ i ∈ Q, k i=1 |λ i | = 1.
By cardinality, there are infinitely many e in (0, 1c k ) with the same collection of λ i . This means that the analytic function λ(x) = λ 0 + k i=1 λ i arccos(c i + x) has an infinity of zeroes in the interval (0, 1c k ), which is absurd. 2

Theorem 5 Every knot is a Chebyshev knot.

Proof. Let K be a knot projecting on the Chebyshev curve {x = T a (t), y = T b (t)}. The crossing points of the projection are obtained for the distinct pairs of values

t = cos k a + h b π, s = cos k a - h b π, k a + h b < 1.
Let us denote these values by (t i , s i ), i = 1 . . . n = 1 2 (a -1)(b -1). By our lemma, let e < 1-cos π ab be a positive number such that the 2n+1 numbers 1, τ i = arccos(t i +e), σ i = arccos(s i + e), i = 1, . . . n, are linearly independent over Q. Let us define the function Z(t) = T c (t + e) (depending on the integer c). We have

Z(t i ) -Z(s i ) = cos c τ i -cos c σ i .
Since the numbers 1, τ i , σ i are linearly independent over Q, the numbers c τ i (mod 2π) and c σ i (mod 2π) are dense in [0, 2π] 2n by Kronecker's theorem. So that we can choose arbitrarily the signs of Z(t i ) -Z(s i ), that is, the over/under nature of the crossing points. 2 . Its Schubert fraction (see [START_REF] Murasugi | Knot Theory and its Applications[END_REF]) is then

-1 + 1 -1 + 1 -1 + 1 -1 + 1 1 + 1 1 + 1 1 = 9 -5 ∼ 9 4 .
This knot is the knot 6 1 .

Example 2: the knot 8 17

The famous 8 17 knot is non reversible and strongly (-)amphicheiral (see [START_REF] Kawauchi | A Survey of Knot Theory[END_REF] p. 128). It is a 3-bridge knot. The Chebyshev curve T 6 (x) = T 5 (y) is one of its diagrams. It can be parametrized by x = T 5 (t), y = T 6 (t), z = T 33 (t+148•10 -4 ). Its reverse can be parametrized by x = T 5 (t), y = T 6 (t), z = T 33 (t -148 • 10 -4 ). We see that it is the reversed image of 8 17 by a half-turn about the y axis.

Conclusion

Let us give a list of the first 2-bridge harmonic knots with their Conway-Rolfsen numbering. Because of their simplicity, we also give their Chebyshev diagrams. A bar over a knot name The 10 159 knot is the harmonic knot H [START_REF] Fischer | Plane Algebraic Curves[END_REF][START_REF] Freudenburg | Curves defined by Chebyshev polynomials[END_REF][START_REF] Koseleff | Chebyshev diagrams for rational knots[END_REF]. Its bridge number is equal to 3. Our Chebyshev parametrization provides an easy proof that it is strongly reversible (compare [START_REF] Kawauchi | A Survey of Knot Theory[END_REF], Appendix F, p. 254).

In conclusion, we have found a great number of distinct harmonic knots. Furthermore, their diagrams have a small number of crossing points. We hope that our Chebyshev models will be useful for the study of knots.

In [START_REF] Koseleff | Chebyshev diagrams for rational knots[END_REF], we classify the harmonic knots H(3, b, c) and H(4, b, c). Even for a = 5, the classification of harmonic knots seems to be a difficult and interesting problem.

Proposition 1

 1 Let a, b be nonnegative integers, a being odd. The affine Chebyshev curve C defined by C : T b (x) -T a (y) = 0 has 1 2 (a -1)(b -1) singular points which are crossing points. These points form two rectangular grids contained in the open square Q

2

 2

Figure 2 :

 2 Figure 2: T 8 (x) = T 3 (y) and its billiard picture

Figure 3 : 7 . 4

 374 Figure 3: s even and s odd for n = 6, 7.

Figure 4 :

 4 Figure 4: The Chebyshev curve T 10 (x) -T 5 (y) = 0 and its billiard picture

Figure 5 :Proposition 5

 55 Figure 5: The knot H(3, 8, 11) is trivial.

Figure 6 :

 6 Figure 6: The right twist and the left twist

Theorem 1 (

 1 Alternate harmonic knots) Let a, b be positive coprime integers, and c = abab. The harmonic diagram H(a, b, c) is alternating. Proof. Using Equation (3), we get z(t)z(s) = -2 sin ch b π sin ck a π = -2 sin (a -1)hπ -ah b π sin (b -1)kπ -bk a π = -2(-1) h(a-1)+k(b-1) sin ah b π sin bk a π . Using Equation (2), we get sign D = -(-1) ah+bk . The crossing points are obtained for the (a -1)(b -1) elements of E = {t u = cos u ab π, 0 ≤ u ≤ ab, a | u, b | u}. Note that t u < t u-1 and that at the crossing point corresponding to t u ∈ E one has sign D = -(-1) u .

Proposition 7

 7 Let a, b be coprime integers. H(a, b, c), H(a, b, 2abc) and H(a, b, 2ab + c) are the same knot.

Proposition 8 2 Corollary 3 HCorollary 4 2 Remark 4

 823424 Let a, b, c be relatively prime integers. There exists c ′ such that H(a, b, c ′ ) is the mirror image of H(a, b, c). Proof. Because a and b are relatively prime, one can write c = αa + βb where α and β are integers. Let us consider c ′ = -αa + βb. We have c ′ ≡ c (mod 2a) and c ′ ≡ -c (mod 2b). For any crossing point of the Chebyshev curve C : T b (x) = T a (y) corresponding to diagrams of both H(a, b, c) and H(a, b, c ′ ), we see that sign D changes to opposite when c is replaced by c ′ . (a, b, ab + ab) is the mirror image of the alternate knot H(a, b, abab). Let a, b be relatively prime integers. There are at most ϕ(a)ϕ(b) different harmonic knots H(a, b, c).Proof. The number of c in[1, ab] that are relatively prime to a and b is ϕ(a)ϕ(b) where ϕ is the Euler function. Because of Propositions 7 and 8, for each c there exists c ′ < ab such that H(a, b, c ′ ) is the mirror image of H(a, b, c). We have at most 1 2 ϕ(a)ϕ(b) different knots when we identify the knots and their mirror images. When a + 1 < b, H(a, b, 1) and its mirror image, H(a, b, a + b) and H(a, b, ba) are trivial.

Proposition 8 :

 8 There is a polynomial automorphism Φ of R 3 such that Φ(H(a, b, c)) = H(a, c ′ , b). They deduce another proof that the harmonic knots H(a, b, a + b) are trivial.

Figure 7 :

 7 Figure 7: The trefoil, the figure-eight knot, and the 6 3 knot.

Figure 8 :

 8 Figure 8: The 6 2 and 9 20 knots .

Figure 9 :Theorem 2

 92 Figure 9: The mirror image of the 10 116 knot.

Figure 10 :

 10 Figure 10: The torus knots 5 1 : H(3, 7, 8) and 7 1 : H(3, 10, 11)

Example 1 : the knot 6 1 D for the 1 2 ( 3 - 1 )Figure 13 :

 1123113 Figure 13: The knot 6 1 is a Chebyshev knot

Figure 14 :

 14 Figure 14: The knot 8 17 and its reverse as Chebyshev knots

  In[START_REF] Koseleff | The first rational Chebyshev knots[END_REF], we give a complete list of Chebyshev parametrizations of the 2-bridge knots of 10 crossings or less. Now, let us give the list of harmonic knots H(5, 6, c). We get 4 = 1 2 ϕ(5)ϕ(6) different types up to mirror symmetry.

	9 1		9 17		9 18
	H(3, 13, 14)	H(3, 11, 16)	H(4, 9, 11)
	9 31		10 37		10 45
	H(3, 10, 17)	H(3, 13, 17)	H(3, 11, 19)
	3 1	4 1		5 1	5 2
	H(3, 4, 5) 0	H(3, 5, 7) 5 2	H(3, 7, 8) 10 159	H(4, 5, 7) 10 116
	H(5, 6, 1)	H(5, 6, 7)	H(5, 6, 13)	H(5, 6, 19)
	6 2	6 3		7 1	7 5
	H(4, 5, 11)	H(3, 7, 11)	H(3, 10, 11)	H(4, 7, 9)
	7 7	8 3		8 7	9 20
	H(3, 8, 13)	H(3, 11, 13)	H(4, 7, 13)	H(4, 7, 17)