Geometry & Topology, Vol. 9 (2005) Paper no. 28, pages 1221--1252.

The nonuniqueness of Chekanov polynomials of Legendrian knots

Paul Melvin, Sumana Shrestha

Abstract. Examples are given of prime Legendrian knots in the standard contact 3-space that have arbitrarily many distinct Chekanov polynomials, refuting a conjecture of Lenny Ng. These are constructed using a new `Legendrian tangle replacement' technique. This technique is then used to show that the phenomenon of multiple Chekanov polynomials is in fact quite common. Finally, building on unpublished work of Yufa and Branson, a tabulation is given of Legendrian fronts, along with their Chekanov polynomials, representing maximal Thurston-Bennequin Legendrian knots for each knot type of nine or fewer crossings. These knots are paired so that the front for the mirror of any knot is obtained in a standard way by rotating the front for the knot.

Keywords. Legendrian knots, contact homology, Chekanov polynomials

AMS subject classification. Primary: 57R17. Secondary: 57M25, 53D12.

E-print: arXiv:math.GT/0411206

DOI: 10.2140/gt.2005.9.1221

Submitted to GT on 10 November 2004. (Revised 3 December 2004.) Paper accepted 4 July 2005. Paper published 24 July 2005.

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Paul Melvin, Sumana Shrestha
Department of Mathematics, Bryn Mawr College
Bryn Mawr, PA 19010, USA

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