×

A new twist on Lorenz links. (English) Zbl 1233.57001

Summary: Twisted torus links are given by twisting a subset of strands on a closed braid representative of a torus link. T-links are a natural generalization given by repeated positive twisting. We establish a one-to-one correspondence between positive braid representatives of Lorenz links and T-links, so Lorenz links and T-links coincide. Using this correspondence, we identify over half of the simplest hyperbolic knots as Lorenz knots. We show that both hyperbolic volume and the Mahler measure of Jones polynomials are bounded for infinite collections of hyperbolic Lorenz links. The correspondence provides unexpected symmetries for both Lorenz links and T-links, and establishes many new results for T-links, including new braid index formulas.

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57M50 General geometric structures on low-dimensional manifolds

Software:

RODES; SnapPea
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Bedient, Classifying 3-trip Lorenz knots, Topology Appl. 20 pp 89– (1985) · Zbl 0576.57007
[2] Birman, Braids: a survey, Handbook of knot theory pp 19– (2005)
[3] Birman, Knotted periodic orbits in dynamical systems-I: Lorenz’s equations, Topology 22 pp 47– (1983) · Zbl 0507.58038
[4] Callahan, The simplest hyperbolic knots, J. Knot Theory Ramifications 3 pp 279– (1999) · Zbl 0933.57010
[5] Callahan, A census of cusped hyperbolic 3-manifolds, Math. Comp. 68 pp 321– (1999) · Zbl 0910.57006
[6] Champanerkar, On the Mahler measure of Jones polynomials under twisting, Algebr. Geom. Topol. 5 pp 1– (2005) · Zbl 1061.57007
[7] Champanerkar, The next simplest hyperbolic knots, J. Knot Theory Ramifications 13 pp 965– (2004) · Zbl 1064.57003
[8] Cromwell, Homogeneous links, J. London Math. Soc. 39 pp 535– (1989) · Zbl 0685.57004
[9] J. Dean Hyperbolic knots with small Seifert-fibered Dehn surgeries PhD Thesis, University of Texas at Austin, May 1996
[10] P. Dehornoy Noeuds de Lorenz Preprint http://www.eleves.ens.fr/home/dehornoy/maths/Lorenz4.pdf
[11] El-Rifai, Necessary and sufficient conditions for Lorenz knots to be closed under satellite construction, Chaos, Solitons Fractals 10 pp 137– (1999) · Zbl 0948.57003
[12] Fiedler, On the degree of the Jones polynomial, Topology 30 pp 1– (1991) · Zbl 0724.57004
[13] Franks, Braids and the Jones polynomial, Trans. Amer. Math. Soc. 303 pp 97– (1987) · Zbl 0647.57002
[14] González-Meneses, The nth root of a braid is unique up to conjugacy, Algebr. Geom. Topol. 3 pp 1103– (2003) · Zbl 1063.20041
[15] E. Ghys Knots and dynamics Preprint, to appear in Proc ICM-2006, Madrid
[16] E. Ghys J. Leys Lorenz and modular flows: a visual introduction Amer. Math. Soc. Feature Column November 2006, available at http://www.ams.org/featurecolumn/archive/lorenz.html
[17] Kawamura, Relations among the lowest degree of the Jones polynomial and geometric invariants for closed positive braids, Comment. Math. Helv. 77 pp 125– (2002) · Zbl 0991.57006
[18] Lorenz, Deterministic non-periodic flow, J. Atmos. Sci. 20 pp 130– (1963) · Zbl 1417.37129
[19] Morton, The Alexander polynomial of a torus knot with twists, J. Knot Theory Ramifications 15 pp 1037– (2006) · Zbl 1115.57004
[20] Murasugi, On the braid index of alternating links, Trans. Amer. Math. Soc. 326 pp 237– (1991) · Zbl 0751.57008
[21] Schubert, Knoten und Vollringe, Acta Math. 90 pp 131– (1953) · Zbl 0051.40403
[22] Stoimenow, Positive knots, closed braids and the Jones polynomial, Ann. Sc. Norm. Super. Pisa Cl. Sci. 2 ((5)) pp 237– (2003) · Zbl 1170.57300
[23] W. Thurston The geometry and topology of three-manifolds electronic 1.1 (2002), available at http://www.msri.org/publications/books/gt3m/
[24] Tucker, A rigorous ODE solver and Smale’s 14th problem, Found. Comput. Math. 2 pp 53– (2002) · Zbl 1047.37012
[25] Viana, What’s new on Lorenz strange attractors, Math. Intelligencer 22 pp 6– (2000) · Zbl 1052.37026
[26] Williams, Lorenz knots are prime, Ergodic Theory Dynam. Systems 4 pp 147– (1983)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.