On the zeroes of the Alexander polynomial of a Lorenz knot
Annales de l'Institut Fourier, Volume 65 (2015) no. 2, pp. 509-548.

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.

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.

Received:
Accepted:
Published online:
DOI: 10.5802/aif.2938
Classification: 57M27,  34C25,  37B40,  37E15,  57M25
Keywords: Lorenz knot, Alexander polynomial, monodromy, surface homeomorphism
@article{AIF_2015__65_2_509_0,
     author = {Dehornoy, Pierre},
     title = {On the zeroes of the {Alexander} polynomial of a {Lorenz} knot},
     journal = {Annales de l'Institut Fourier},
     pages = {509--548},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {65},
     number = {2},
     year = {2015},
     doi = {10.5802/aif.2938},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2938/}
}
TY  - JOUR
TI  - On the zeroes of the Alexander polynomial of a Lorenz knot
JO  - Annales de l'Institut Fourier
PY  - 2015
DA  - 2015///
SP  - 509
EP  - 548
VL  - 65
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2938/
UR  - https://doi.org/10.5802/aif.2938
DO  - 10.5802/aif.2938
LA  - en
ID  - AIF_2015__65_2_509_0
ER  - 
%0 Journal Article
%T On the zeroes of the Alexander polynomial of a Lorenz knot
%J Annales de l'Institut Fourier
%D 2015
%P 509-548
%V 65
%N 2
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.2938
%R 10.5802/aif.2938
%G en
%F AIF_2015__65_2_509_0
Dehornoy, Pierre. On the zeroes of the Alexander polynomial of a Lorenz knot. Annales de l'Institut Fourier, Volume 65 (2015) no. 2, pp. 509-548. doi : 10.5802/aif.2938. https://aif.centre-mersenne.org/articles/10.5802/aif.2938/

[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, Mathematical Notes, Tome 48, Princeton University Press, Princeton, NJ, 2012, xvi+254 pages (Translated from the 1979 French original by Djun M. Kim and Dan Margalit) | MR: 3053012 | Zbl: 1244.57005

[10] Fomenko, A. T. Symplectic geometry, Advanced Studies in Contemporary Mathematics, Tome 5, Gordon and Breach Publishers, Luxembourg, 1995, xvi+467 pages (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) (Contemp. Math.) Tome 20, Amer. Math. Soc., Providence, RI, 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, Lecture Notes in Mathematics, Tome 1654, Springer-Verlag, Berlin, 1997, x+208 pages | 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 (Prog. Math. Phys.) Tome 66, Birkhäuser/Springer, Basel, 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, xxii+420 pages (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

Cited by Sources: