Legendrian and transverse twist knots

Abstract

In 1997, Chekanov gave the first example of a Legendrian nonsimple knot type: the m.52/ knot. Epstein, Fuchs, and Meyer extended his result by showing that there are at least n different Legendrian representatives with maximal Thurston-Bennequin number of the twist knot K+2n with crossing number 2n C 1. In this paper we give a complete classification of Legendrian and transverse representatives of twist knots. In particular, we show that K -2n has exactly dn2=2e Legendrian representatives with maximal Thurston-Bennequin number, and dn=2e transverse representatives with maximal self-linking number. Our techniques include convex surface theory, Legendrian ruling invariants, and Heegaard Floer homology. © European Mathematical Society 2013.