The nonuniqueness of Chekanov polynomials of Legendrian knots

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 Finally, building on unpublished work of Yufa and Branson, a tabulation is given of Legendrian fronts, along with their Chekanov polynomials, representing maximal Thurston-Bennequin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

Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol9/paper28.abs.html

References in corpus (1)

• Duality for Legendrian contact homology

Cited by in corpus (14)

• A Duality Exact Sequence for Legendrian Contact Homology
• Knot and braid invariants from contact homology II, with an appendix written jointly with Siddhartha Gadgil
• Knot and braid invariants from contact homology I
• Generating families and Legendrian contact homology in the standard contact space
• An atlas of Legendrian knots
• Lagrangian Cobordisms via Generating Families: Constructions and Geography
• Duality for Legendrian contact homology
• Augmentations are Sheaves
• Obstructions to Lagrangian Cobordisms between Legendrians via Generating Families
• The contact homology of Legendrian knots with maximal Thurston-Bennequin invariant
• The cardinality of the augmentation category of a Legendrian link
• Topologically Distinct Lagrangian and Symplectic Fillings
• Product Structures for Legendrian Contact Homology
• Geography of bilinearized Legendrian contact homology