Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On Legendrian knots and polynomial invariants


Author: Emmanuel Ferrand
Journal: Proc. Amer. Math. Soc. 130 (2002), 1169-1176
MSC (1991): Primary 53C15, 57M25
DOI: https://doi.org/10.1090/S0002-9939-01-06153-6
Published electronically: September 14, 2001
MathSciNet review: 1873793
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [AG] Arnold, V., Givental, A. : Symplectic geometry, Encyclopedia of Mathematical Science vol. 4, Springer-Verlag.
  • [Be] D. Bennequin : Entrelacements et équation de Pfaff. Astérisque 107-108 (1983), 83-161. MR 86e:58070
  • [CG] Chmutov, S., Goryunov, V. : Polynomial invariants of Legendrian links and wave fronts, KNOTS '96 (Tokyo), World Sci. Publishing, River Edge, NJ, 1997, 239-256. MR 99k:57010
  • [CGM] Chmutov, S., Goryunov, V., Murakami, H. : Regular Legendrian knots and the HOMFLY polynomial of immersed plane curves, Math. Ann. 317 (2000), 389-413. CMP 2000:16
  • [Cr] Cromwell, P.R. : Homogeneous links J. London Math. Soc. (2) 39 (1989), no. 3, 535-552. MR 90f:57001
  • [DM] Dasbach, O., Mangum, B.S. : On McMullen's and other inequalities for the Thurston norm of link complements, preprint, 1999.
  • [EH] Etnyre, J. B., Honda, K. : Knots and contact geometry, arXiv:math.GT/006112.
  • [Ep] Epstein, J. : On the Invariants and Isotopies of Legendrian and Transverse Knots, dissertation, U. C. Davis, (1997).
  • [Fe] Ferrand, E. : On Legendre cobordisms Amer. Math. Soc. Transl. (2) Vol. 190, (1999), 23-35. CMP 2000:08
  • [FW] Franks, J., Williams, R. F. : Braids and the Jones polynomial Trans. Amer. Math. Soc. 303 (1987), no. 1, 97-108. MR 88k:57006
  • [GH] Goryunov, V. V., Hill, J. W. : A Bennequin number estimate for transverse knots, Singularity theory (Liverpool, 1996), London Math. Soc. Lecture Note Ser., 263, (1999), 265-280. MR 2000f:57006
  • [FT] Fuchs, D., Tabachnikov, S. : Invariants of Legendrian and transverse knots in the standard contact space, Topology 36 (1997), no. 5, 1025-1053. MR 99a:57006
  • [HT] Hoste, J., Thistlethwaite, M. : Knotscape, http://www.math.utk.edu/morwen/knotscape. html
  • [Ka] Kanda, Y. : On the Thurston-Bennequin invariant of Legendrian knots and non exacteness of the Bennequin inequality, Invent. Math. 133 (1998), no. 2, 227-242. MR 99g:57030
  • [Kau] Kauffman, L. : Knots and Physics, World Scientific, 1991. MR 93b:57010
  • [Mo] Morton, H. : Seifert circles and knot polynomials, Math. Proc. Cambridge Philos. Soc. 99 (1986), no. 1, 107-109. MR 87c:57006
  • [Ng] Ng, L. : Maximal Thurston-Bennequin number of two-bridge knots, preprint.
  • [Ru1] Rudolph, L. : Construction of quasipositive knots and links, II, Contemp. Math. 35, 485-491. MR 86f:57005
  • [Ru2] Rudolph, L. : A congruence between link polynomials, Math. Proc. Camb. Phil. Soc. (1990) 107 319-327. MR 90k:57010
  • [Ru3] Rudolph, L. : Quasi-positive annuli (construction of quasipositive knots and links, IV), J. Knot Th. Ramifications (1992) 1, 4, 461-466. MR 94c:57017
  • [Ru4] Rudolph, L. : Totally tangential links of intersection of complex curves with round spheres, Topology'90 (eds. Apanasov et al.), DeGryuter, 1992, 343-349. MR 94d:57027
  • [Ru5] Rudolph, L. : Quasipositivity as an obstruction to sliceness, Bull. A.M.S. (1993) 29, 1, 51-59. MR 94d:57028
  • [Ru6] Rudolph, L. : An obstruction to sliceness via contact geometry and ``classical'' gauge theory, Invent. Math. 119 (1995), 155-163. MR 95k:57013
  • [Ta] Tabachnikov, S. : Estimates for the Bennequin number of Legendrian links from state models for knot polynomials, Math. Res. Lett. 4 (1997), no. 1, 143-156. MR 98k:57023
  • [Tan] Tanaka, T. : Maximal Bennequin numbers and Kauffman polynomials of positive links, Proc. Amer. Math. Soc. 127 (1999), no. 11, 3427-3432. MR 2000b:57014
  • [Yo] Yokota, Y. : Polynomial invariants of positive links, Topology 31 (1992), no. 4, 805-811. MR 93k:57028

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 53C15, 57M25

Retrieve articles in all journals with MSC (1991): 53C15, 57M25


Additional Information

Emmanuel Ferrand
Affiliation: Institut Fourier, BP 74, 38402 St Martin d’Hères Cedex, France
Email: emmanuel.ferrand@ujf-grenoble.fr

DOI: https://doi.org/10.1090/S0002-9939-01-06153-6
Keywords: Contact topology, polynomial invariants of knots
Received by editor(s): July 11, 2000
Received by editor(s) in revised form: October 24, 2000
Published electronically: September 14, 2001
Communicated by: Ronald A. Fintushel
Article copyright: © Copyright 2001 American Mathematical Society

American Mathematical Society