# Legendrian and transverse twist knots

### Lenhard L. Ng

Duke University, Durham, USA### John B. Etnyre

Georgia Institute of Technology, Atlanta, USA### Vera Vértesi

Massachusetts Institute of Technology, Cambridge, USA

## Abstract

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 with maximal Thurston--Bennequin number of the twist knot $K_{-2n}$ with crossing number $2n+1$. 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 $\lceil\frac{n^2}2\rceil$ 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.

## Cite this article

Lenhard L. Ng, John B. Etnyre, Vera Vértesi, Legendrian and transverse twist knots. J. Eur. Math. Soc. 15 (2013), no. 3, pp. 969–995

DOI 10.4171/JEMS/383