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


In 1997, Chekanov gave the first example of a Legendrian nonsimple knot type: the m(52)m(5_2) knot. Epstein, Fuchs, and Meyer extended his result by showing that there are at least nn different Legendrian representatives with maximal Thurston--Bennequin number of the twist knot K2nK_{-2n} with crossing number 2n+12n+1. In this paper we give a complete classification of Legendrian and transverse representatives of twist knots. In particular, we show that K2nK_{-2n} has exactly n22\lceil\frac{n^2}2\rceil Legendrian representatives with maximal Thurston–Bennequin number, and n2\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