Geometry & Topology

The nonuniqueness of Chekanov polynomials of Legendrian knots

Paul Melvin and Sumana Shrestha

Full-text: Open access

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 Finally, building on unpublished work of Yufa and Branson, a tabulation is given of Legendrian fronts, along with their Chekanov polynomials, representing maximal Thurston–Bennequin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

Article information

Source
Geom. Topol., Volume 9, Number 3 (2005), 1221-1252.

Dates
Received: 10 November 2004
Revised: 3 December 2004
Accepted: 4 July 2005
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513799635

Digital Object Identifier
doi:10.2140/gt.2005.9.1221

Mathematical Reviews number (MathSciNet)
MR2174265

Zentralblatt MATH identifier
1084.57009

Subjects
Primary: 57R17: Symplectic and contact topology
Secondary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 53D12: Lagrangian submanifolds; Maslov index

Keywords
Legendrian knots contact homology Chekanov polynomials

Citation

Melvin, Paul; Shrestha, Sumana. The nonuniqueness of Chekanov polynomials of Legendrian knots. Geom. Topol. 9 (2005), no. 3, 1221--1252. doi:10.2140/gt.2005.9.1221. https://projecteuclid.org/euclid.gt/1513799635


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