On the zeroes of the Alexander polynomial of a Lorenz knot
[Sur les racines du polynome d’Alexander d’un nœud de Lorenz]
Dehornoy, Pierre
Annales de l'Institut Fourier, Tome 65 (2015), p. 509-548 / Harvested from Numdam
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On montre que les racines du polynome d’Alexander d’un nœud de Lorenz sont situées dans un anneau dont l’épaisseur dépend explicitement du genre et de l’indice de tresse du nœud considéré.

We show that the zeroes of the Alexander polynomial of a Lorenz knot all lie in some annulus whose width depends explicitly on the genus and the braid index of the considered knot.

Publié le : 2015-01-01
DOI : https://doi.org/10.5802/aif.2938
Classification:  57M27,  34C25,  37B40,  37E15,  57M25
Mots clés: Nœud de Lorenz, polynome d’Alexander, monodromie, homéomorphisme de surface 
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     author = {Dehornoy, Pierre},
     title = {On the zeroes of the Alexander polynomial of a Lorenz knot},
     journal = {Annales de l'Institut Fourier},
     volume = {65},
     year = {2015},
     pages = {509-548},
     doi = {10.5802/aif.2938},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2015__65_2_509_0}
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Dehornoy, Pierre. On the zeroes of the Alexander polynomial of a Lorenz knot. Annales de l'Institut Fourier, Tome 65 (2015) pp. 509-548. doi : 10.5802/aif.2938. http://gdmltest.u-ga.fr/item/AIF_2015__65_2_509_0/

[1] ArnolʼD, V. I. The asymptotic Hopf invariant and its applications, Selecta Math. Soviet., Tome 5 (1986) no. 4, pp. 327-345 (Selected translations) | MR 891881 | Zbl 0623.57016

[2] Baader, Sebastian; Marché, Julien Asymptotic Vassiliev invariants for vector fields, Bull. Soc. Math. France, Tome 140 (2012) no. 4, p. 569-582 (2013) http://arxiv.org/abs/0810.3870 | Numdam | MR 3059851 | Zbl 1278.57017

[3] Birman, Joan; Brinkmann, Peter; Kawamuro, Keiko Polynomial invariants of pseudo-Anosov maps, J. Topol. Anal., Tome 4 (2012) no. 1, pp. 13-47 | Article | MR 2914872 | Zbl 1268.57002

[4] Birman, Joan; Kofman, Ilya A new twist on Lorenz links, J. Topol., Tome 2 (2009) no. 2, pp. 227-248 | Article | MR 2529294 | Zbl 1233.57001

[5] Birman, Joan S.; Williams, R. F. Knotted periodic orbits in dynamical systems. I. Lorenz’s equations, Topology, Tome 22 (1983) no. 1, pp. 47-82 (Erratum: http://www.math.columbia.edu/~jb/bw-KPO-I-erratum.pdf (2006)) | Article | MR 682059 | Zbl 0507.58038

[6] Dehornoy, Pi. Atlas of Lorenz knots (http://www-fourier.ujf-grenoble.fr/~dehornop/maths/atlaslorenz.txt)

[7] Dehornoy, Pierre Les nœuds de Lorenz, Enseign. Math. (2), Tome 57 (2011) no. 3-4, pp. 211-280 | Article | MR 2920729 | Zbl 1244.57012

[8] Farb, B.; Margalit, D. A Primer on Mapping Class Groups (To be published by Princeton Univ. Press.)

[9] Fathi, Albert; Laudenbach, François; Poénaru, Valentin Thurston’s work on surfaces, Princeton University Press, Princeton, NJ, Mathematical Notes, Tome 48 (2012), pp. xvi+254 (Translated from the 1979 French original by Djun M. Kim and Dan Margalit) | MR 3053012 | Zbl 1244.57005

[10] Fomenko, A. T. Symplectic geometry, Gordon and Breach Publishers, Luxembourg, Advanced Studies in Contemporary Mathematics, Tome 5 (1995), pp. xvi+467 (Translated from the 1988 Russian original by R. S. Wadhwa) | MR 1673400 | Zbl 0873.58031

[11] Franks, John; Williams, R. F. Braids and the Jones polynomial, Trans. Amer. Math. Soc., Tome 303 (1987) no. 1, pp. 97-108 | Article | MR 896009 | Zbl 0647.57002

[12] Gabai, David The Murasugi sum is a natural geometric operation, Low-dimensional topology (San Francisco, Calif., 1981), Amer. Math. Soc., Providence, RI (Contemp. Math.) Tome 20 (1983), pp. 131-143 | Article | MR 718138 | Zbl 0584.57003

[13] Gabai, David Detecting fibred links in S 3 , Comment. Math. Helv., Tome 61 (1986) no. 4, pp. 519-555 | Article | MR 870705 | Zbl 0621.57003

[14] Gambaudo, Jean-Marc; Ghys, Étienne Signature asymptotique d’un champ de vecteurs en dimension 3, Duke Math. J., Tome 106 (2001) no. 1, pp. 41-79 | Article | MR 1810366 | Zbl 1010.37010

[15] Ghrist, Robert W.; Holmes, Philip J.; Sullivan, Michael C. Knots and links in three-dimensional flows, Springer-Verlag, Berlin, Lecture Notes in Mathematics, Tome 1654 (1997), pp. x+208 | MR 1480169 | Zbl 0869.58044

[16] Ghys, Étienne Knots and dynamics, International Congress of Mathematicians. Vol. I, Eur. Math. Soc., Zürich (2007), pp. 247-277 | Article | MR 2334193 | Zbl 1125.37032

[17] Ghys, Étienne The Lorenz attractor, a paradigm for chaos, Chaos, Birkhäuser/Springer, Basel (Prog. Math. Phys.) Tome 66 (2013), pp. 1-54 | Article | MR 3204181

[18] Hironaka, Eriko Small dilatation mapping classes coming from the simplest hyperbolic braid, Algebr. Geom. Topol., Tome 10 (2010) no. 4, pp. 2041-2060 | Article | MR 2728483 | Zbl 1221.57028

[19] Hironaka, Eriko; Kin, Eiko A family of pseudo-Anosov braids with small dilatation, Algebr. Geom. Topol., Tome 6 (2006), p. 699-738 (electronic) | Article | MR 2240913 | Zbl 1126.37014

[20] Kawauchi, Akio A survey of knot theory, Birkhäuser Verlag, Basel (1996), pp. xxii+420 (Translated and revised from the 1990 Japanese original by the author) | MR 1417494 | Zbl 0861.57001

[21] Lanneau, Erwan; Thiffeault, Jean-Luc On the minimum dilatation of pseudo-Anosov homeromorphisms on surfaces of small genus, Ann. Inst. Fourier (Grenoble), Tome 61 (2011) no. 1, pp. 105-144 | Article | Numdam | MR 2828128 | Zbl 1237.37027

[22] Livingstone, Ch. Table of knots invariants (http://www.indiana.edu/~knotinfo/)

[23] Lorenz, E.N. Deterministic Nonperiodic Flow, J. Atmos. Sci., Tome 20 (1963) no. 2, pp. 130-141 | Article

[24] Milnor, John W. Infinite cyclic coverings, Conference on the Topology of Manifolds (Michigan State Univ., E. Lansing, Mich., 1967), Prindle, Weber & Schmidt, Boston, Mass. (1968), pp. 115-133 | MR 242163 | Zbl 0179.52302

[25] Murasugi, Kunio On the genus of the alternating knot. I, II, J. Math. Soc. Japan, Tome 10 (1958), p. 94-105, 235–248 | Article | MR 99664 | Zbl 0106.16701

[26] Penner, R. C. Bounds on least dilatations, Proc. Amer. Math. Soc., Tome 113 (1991) no. 2, pp. 443-450 | Article | MR 1068128 | Zbl 0726.57013

[27] Thurston, William P. On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.), Tome 19 (1988) no. 2, pp. 417-431 | Article | MR 956596 | Zbl 0674.57008