On Legendrian knots and polynomial invariants

Ferrand, Emmanuel

doi:10.1090/S0002-9939-01-06153-6

On Legendrian knots and polynomial invariants
HTML articles powered by AMS MathViewer

by Emmanuel Ferrand PDF

Proc. Amer. Math. Soc. 130 (2002), 1169-1176 Request permission

Abstract:

It is proved in this note that the analogues of the Bennequin inequality which provide an upper bound for the Bennequin invariant of a Legendrian knot in the standard contact three dimensional space in terms of the least degree in the framing variable of the HOMFLY and the Kauffman polynomials are not sharp. Furthermore, the relationships between these restrictions on the range of the Bennequin invariant are investigated, which leads to a new simple proof of the inequality involving the Kauffman polynomial.

References

Arnold, V., Givental, A. : Symplectic geometry, Encyclopedia of Mathematical Science vol. 4, Springer-Verlag.
Daniel Bennequin, Entrelacements et équations de Pfaff, Third Schnepfenried geometry conference, Vol. 1 (Schnepfenried, 1982) Astérisque, vol. 107, Soc. Math. France, Paris, 1983, pp. 87–161 (French). MR 753131
S. Chmutov and V. Goryunov, Polynomial invariants of Legendrian links and their fronts, KNOTS ’96 (Tokyo), World Sci. Publ., River Edge, NJ, 1997, pp. 239–256. MR 1664965
Chmutov, S., Goryunov, V., Murakami, H. : Regular Legendrian knots and the HOMFLY polynomial of immersed plane curves, Math. Ann. 317 (2000), 389–413.
P. R. Cromwell, Homogeneous links, J. London Math. Soc. (2) 39 (1989), no. 3, 535–552. MR 1002465, DOI 10.1112/jlms/s2-39.3.535
Dasbach, O., Mangum, B.S. : On McMullen’s and other inequalities for the Thurston norm of link complements, preprint, 1999.
Etnyre, J. B., Honda, K. : Knots and contact geometry, arXiv:math.GT/006112.
Epstein, J. : On the Invariants and Isotopies of Legendrian and Transverse Knots, dissertation, U. C. Davis, (1997).
Ferrand, E. : On Legendre cobordisms Amer. Math. Soc. Transl. (2) Vol. 190, (1999), 23-35.
John Franks and R. F. Williams, Braids and the Jones polynomial, Trans. Amer. Math. Soc. 303 (1987), no. 1, 97–108. MR 896009, DOI 10.1090/S0002-9947-1987-0896009-2
V. V. Goryunov and J. W. Hill, A Bennequin number estimate for transverse knots, Singularity theory (Liverpool, 1996) London Math. Soc. Lecture Note Ser., vol. 263, Cambridge Univ. Press, Cambridge, 1999, pp. xx, 265–280. MR 1709357
Dmitry Fuchs and Serge Tabachnikov, Invariants of Legendrian and transverse knots in the standard contact space, Topology 36 (1997), no. 5, 1025–1053. MR 1445553, DOI 10.1016/S0040-9383(96)00035-3
Hoste, J., Thistlethwaite, M. : Knotscape, http://www.math.utk.edu/~morwen/knotscape.html
Yutaka Kanda, On the Thurston-Bennequin invariant of Legendrian knots and nonexactness of Bennequin’s inequality, Invent. Math. 133 (1998), no. 2, 227–242. MR 1632790, DOI 10.1007/s002220050245
Louis H. Kauffman, Knots and physics, Series on Knots and Everything, vol. 1, World Scientific Publishing Co., Inc., River Edge, NJ, 1991. MR 1141156, DOI 10.1142/9789812796226
H. R. Morton, Seifert circles and knot polynomials, Math. Proc. Cambridge Philos. Soc. 99 (1986), no. 1, 107–109. MR 809504, DOI 10.1017/S0305004100063982
Ng, L. : Maximal Thurston-Bennequin number of two-bridge knots, preprint.
Lee Rudolph, Constructions of quasipositive knots and links. II, Four-manifold theory (Durham, N.H., 1982) Contemp. Math., vol. 35, Amer. Math. Soc., Providence, RI, 1984, pp. 485–491. MR 780596, DOI 10.1090/conm/035/780596
Lee Rudolph, A congruence between link polynomials, Math. Proc. Cambridge Philos. Soc. 107 (1990), no. 2, 319–327. MR 1027784, DOI 10.1017/S0305004100068584
Lee Rudolph, Quasipositive annuli. (Constructions of quasipositive knots and links. IV), J. Knot Theory Ramifications 1 (1992), no. 4, 451–466. MR 1194997, DOI 10.1142/S0218216592000227
Lee Rudolph, Totally tangential links of intersection of complex plane curves with round spheres, Topology ’90 (Columbus, OH, 1990) Ohio State Univ. Math. Res. Inst. Publ., vol. 1, de Gruyter, Berlin, 1992, pp. 343–349. MR 1184419
Lee Rudolph, Quasipositivity as an obstruction to sliceness, Bull. Amer. Math. Soc. (N.S.) 29 (1993), no. 1, 51–59. MR 1193540, DOI 10.1090/S0273-0979-1993-00397-5
Lee Rudolph, An obstruction to sliceness via contact geometry and “classical” gauge theory, Invent. Math. 119 (1995), no. 1, 155–163. MR 1309974, DOI 10.1007/BF01245177
Serge Tabachnikov, Estimates for the Bennequin number of Legendrian links from state models for knot polynomials, Math. Res. Lett. 4 (1997), no. 1, 143–156. MR 1432817, DOI 10.4310/MRL.1997.v4.n1.a13
Toshifumi Tanaka, Maximal Bennequin numbers and Kauffman polynomials of positive links, Proc. Amer. Math. Soc. 127 (1999), no. 11, 3427–3432. MR 1616601, DOI 10.1090/S0002-9939-99-04983-7
Yoshiyuki Yokota, Polynomial invariants of positive links, Topology 31 (1992), no. 4, 805–811. MR 1191382, DOI 10.1016/0040-9383(92)90011-6