The nonuniqueness of Chekanov polynomials of Legendrian knots

Melvin, Paul; Shrestha, Sumana

doi:10.2140/gt.2005.9.1221

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The nonuniqueness of Chekanov polynomials of Legendrian knots

Paul Melvin and Sumana Shrestha

Geometry & Topology 9 (2005) 1221–1252

DOI: 10.2140/gt.2005.9.1221

arXiv: math.GT/0411206

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 Finally, building on unpublished work of Yufa and Branson, a tabulation is given of Legendrian fronts, along with their Chekanov polynomials, representing maximal Thurston–Bennequin Legendrian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

Mathematical Subject Classification 2000

Primary: 57R17

Secondary: 57M25, 53D12

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Publication

Received: 10 November 2004

Revised: 3 December 2004

Accepted: 4 July 2005

Published: 24 July 2005

Proposed: Yasha Eliashberg

Seconded: Robion Kirby, Joan Birman

Authors

Paul Melvin
Department of Mathematics Bryn Mawr College Bryn Mawr Pennsylvania 19010 USA

Sumana Shrestha

Volume 9, issue 3 (2005)