Geometry & Topology

Generating function polynomials for legendrian links

Lisa Traynor

Full-text: Open access

Abstract

It is shown that, in the 1–jet space of the circle, the swapping and the flyping procedures, which produce topologically equivalent links, can produce nonequivalent legendrian links. Each component of the links considered is legendrian isotopic to the 1–jet of the 0–function, and thus cannot be distinguished by the classical rotation number or Thurston–Bennequin invariants. The links are distinguished by calculating invariant polynomials defined via homology groups associated to the links through the theory of generating functions. The many calculations of these generating function polynomials support the belief that these polynomials carry the same information as a refined version of Chekanov’s first order polynomials which are defined via the theory of holomorphic curves.

Article information

Source
Geom. Topol., Volume 5, Number 2 (2001), 719-760.

Dates
Received: 15 June 2001
Revised: 6 September 2001
Accepted: 5 October 2001
First available in Project Euclid: 21 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513883042

Digital Object Identifier
doi:10.2140/gt.2001.5.719

Mathematical Reviews number (MathSciNet)
MR1871403

Zentralblatt MATH identifier
1030.53086

Subjects
Primary: 53D35: Global theory of symplectic and contact manifolds [See also 57Rxx]
Secondary: 58E05: Abstract critical point theory (Morse theory, Ljusternik-Schnirelman (Lyusternik-Shnirel m an) theory, etc.)

Keywords
contact topology contact homology generating functions legendrian links knot polynomials

Citation

Traynor, Lisa. Generating function polynomials for legendrian links. Geom. Topol. 5 (2001), no. 2, 719--760. doi:10.2140/gt.2001.5.719. https://projecteuclid.org/euclid.gt/1513883042


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References

  • C Adams, The Knot Book, W H Freeman and Company (1994) \ref
  • D Bennequin, Entrelacements et équation de Pfaff , Astérique, 107–108 (1983), 87–162 \ref
  • M Chaperon, On generating families, The Floer Memorial Volume (H Hofer, C Taubes, A Weinstein, E Zehnder, eds.), Birkhäuser (1995) \ref
  • Yu Chekanov, Critical points of quasifunctions, and generating families of legendrian manifolds , Funktsional.-Anal.-i Prilozhen (in Russian), 30 (1996), 56–69 \ref
  • Yu Chekanov, Differential algebras of Legendrian links , Fields Institute preprint FI-ST1997-010 (1997) \ref
  • J H Conway, An enumeration of knots and links, and some of their algebraic properties, Computational problems in Abstract Algebra, Proc. Conf. Oxford, 1967 (J Leech, ed.), Pergamon Press, New York (1970) \ref
  • Y Eliashberg, Legendrian and transversal knots in tight contact 3–manifolds, Topological Methods in Modern Mathematics (L R Goldberg and A V Phillips, eds.), Publish or Perish, Inc. (1993) \ref
  • Y Eliashberg, Invariants in contact topology, Doc. Math. J. DMV, Extra Volume ICM 1998, (electronic), 327–338 \ref
  • Y Eliashberg
  • M Gromov, Lagrangian intersections theory: finite-dimensional approach, Geometry of differential equations, Amer. Math. Soc. Transl. Ser. 2, 186, Amer. Math. Soc. Providence, RI (1998) \ref
  • Y Eliashberg
  • A Givental
  • H Hofer, Symplectic Field Theory , e-print\nl arXiv:math.SG/0010059 \ref
  • E Ferrand, On Legendrian knots and polynomial invariants , e-print arXiv:\nl math.GT/0002250 \ref
  • R Guy \bookUnsolved Problems in Number Theory, Springer–Verlag (1991) \ref
  • L Ng \bookInvariants of Legendrian Links, MIT Ph.D. Thesis (2001) \ref
  • L Ng
  • L Traynor, in preparation \ref
  • S Tabachnikov, Estimates for the Bennequin number of Legendrian links from state models for knot polynomials , Math. Res. Lett., 4 (1997), 143–156 \ref
  • D Théret, A complete proof of Viterbo's uniqueness theorem on generating functions , Topology Appl., 96 (1999), 249–266 \ref
  • D Théret, A Lagrangian camel , Comm. Math. Helv., 74 (1999), 591–614 \ref
  • L Traynor, Symplectic homology via generating functions , Geom. Funct. Anal., 4 (1994), 718–748 \ref
  • L Traynor, Legendrian circular helix links , Math. Proc. Camb. Phil. Soc., 122 (1997), 301–314 \ref
  • L Traynor, A legendrian stratification of rational tangles , Journal of Knot Theory and its Ramifications, 7 (1998), 659–700 \ref
  • C Viterbo, Symplectic topology as the geometry of generating functions , Math. Ann., 292 (1992), 685–710 \ref
  • C Viterbo, Generating Functions, Symplectic Geometry, and Applications, Proceedings of the International Congress of Mathematicians, Zürich, Switzerland 1994, Birkhäuser–Verlag, Basel, Switzerland (1995), 537–547 s