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The nonuniqueness of Chekanov polynomials of Legendrian knots

Paul Melvin and Sumana Shrestha

Geometry & Topology 9 (2005) 1221–1252

arXiv: math.GT/0411206


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 Finally, building on unpublished work of Yufa and Branson, a tabulation is given of Legendrian fronts, along with their Chekanov polynomials, representing maximal Thurston–Bennequin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

Legendrian knots, contact homology, Chekanov polynomials
Mathematical Subject Classification 2000
Primary: 57R17
Secondary: 57M25, 53D12
Forward citations
Received: 10 November 2004
Revised: 3 December 2004
Accepted: 4 July 2005
Published: 24 July 2005
Proposed: Yasha Eliashberg
Seconded: Robion Kirby, Joan Birman
Paul Melvin
Department of Mathematics
Bryn Mawr College
Bryn Mawr
Pennsylvania 19010
Sumana Shrestha