Legendrian and transverse twist knots
Joint work with Lenhard Ng and Vera V\'ertesi,
J.\ European Math.\ Soc. 15 (2013), 969--995.

In 1997, Chekanov gave the first example of a Legendrian nonsimple knot type: the $m(5_2)$ knot. Epstein, Fuchs, and Meyer extended his result by showing that there are at least $n$ different Legendrian representatives of the $m((2n+1)_2)$ knot with maximal Thurston--Bennequin number. In this paper we give a complete classification of Legendrian and transverse representatives of twist knots. In particular, we show that the $m((2n+1)_2)$ knot has exactly $\lceil\frac{n^2}2\rceil$ Legendrian representatives with maximal Thurston--Bennequin number, and $\lceil\frac{n}{2}\rceil$ transverse representatives with maximal self-linking number. Our techniques include convex surface theory, Legendrian ruling invariants, and Heegaard Floer homology.

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