×

Maximal Thurston-Bennequin number of two-bridge links. (English) Zbl 1056.57010

The author determines the maximal Thurston-Bennequin number for a two-bridge knot or oriented two-bridge link in the standard contact structure, by showing that the upper bound in terms of the Kauffman polynomial due to L. Rudolf [Math. Proc. Camb. Philos. Soc. 107, No. 2, 319–327 (1990; Zbl 0703.57005)] and S. Tabachnikov [Math. Res. Lett. 4, No. 1, 143–156 (1997; Zbl 0877.57001)] is sharp. As an application, he determines the maximal Thurston-Bennequin number for prime knots with \(\leq 9\) crossings, except the mirror image of the \(9_{42}\) knot in Rolfsen’s table.

MSC:

57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57M15 Relations of low-dimensional topology with graph theory
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57R17 Symplectic and contact topology in high or arbitrary dimension
PDF BibTeX XML Cite
Full Text: DOI arXiv EuDML EMIS

References:

[1] D Bennequin, Entrelacements et équations de Pfaff, Astérisque 107, Soc. Math. France (1983) 87 · Zbl 0573.58022
[2] G Burde, H Zieschang, Knots, de Gruyter Studies in Mathematics 5, Walter de Gruyter & Co. (1985) · Zbl 0568.57001
[3] J Epstein, On the invariants and isotopies of Legendrian and transverse knots, PhD thesis, UC Davis (1997)
[4] J B Etnyre, K Honda, Knots and contact geometry I: Torus knots and the figure eight knot, J. Symplectic Geom. 1 (2001) 63 · Zbl 1037.57021
[5] E Ferrand, On Legendrian knots and polynomial invariants, Proc. Amer. Math. Soc. 130 (2002) 1169 · Zbl 1005.53023
[6] D Fuchs, S Tabachnikov, Invariants of Legendrian and transverse knots in the standard contact space, Topology 36 (1997) 1025 · Zbl 0904.57006
[7] L H Kauffman, On knots, Annals of Mathematics Studies 115, Princeton University Press (1987) · Zbl 0627.57002
[8] W B R Lickorish, Linear skein theory and link polynomials, Topology Appl. 27 (1987) 265 · Zbl 0638.57005
[9] K Murasugi, Knot theory and its applications, Birkhäuser (1996) · Zbl 0864.57001
[10] L Ng, Invariants of Legendrian links, PhD thesis, MIT (2001)
[11] D Rolfsen, Knots and links, Mathematics Lecture Series 7, Publish or Perish (1990) · Zbl 0854.57002
[12] L Rudolph, A congruence between link polynomials, Math. Proc. Cambridge Philos. Soc. 107 (1990) 319 · Zbl 0703.57005
[13] H Schubert, Knoten mit zwei Brücken, Math. Z. 65 (1956) 133 · Zbl 0071.39002
[14] S Tabachnikov, Estimates for the Bennequin number of Legendrian links from state models for knot polynomials, Math. Res. Lett. 4 (1997) 143 · Zbl 0877.57001
[15] T Tanaka, Maximal Bennequin numbers and Kauffman polynomials of positive links, Proc. Amer. Math. Soc. 127 (1999) 3427 · Zbl 0983.57008
[16] N Yufa, Thurston-Bennequin invariant of Legendrian knots, senior thesis, MIT (2001)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.