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 Finally, building on unpublished work of Yufa and Branson, a tabulation is given of Legendrian fronts, along with their Chekanov polynomials, representing maximal Thurston–Bennequin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

Primary: 57R17

Secondary: 53D12, 57M25

References

Forward citations

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