The nonuniqueness of Chekanov polynomials of Legendrian knots

Details The nonuniqueness of Chekanov polynomials of Legendrian knots The nonuniqueness of Chekanov polynomials of Legendrian knots

2004-11-09

arxiv.org/abs/math/0411206v3

Geom. Topol. 9 (2005) 1221-1252

Mathematics

Geometric Topology

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

Scientific paper

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 Finally, building on unpublished work of Yufa and Branson, a tabulation is given of Legendrian fronts, along with their Chekanov polynomials, representing maximal Thurston-Bennequin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

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