In 1997, Chekanov gave the first example of a Legendrian nonsimple knot type: the knot. Epstein, Fuchs, and Meyer extended his result by showing that there are at least different Legendrian representatives with maximal Thurston--Bennequin number of the twist knot with crossing number . In this paper we give a complete classification of Legendrian and transverse representatives of twist knots. In particular, we show that has exactly Legendrian representatives with maximal Thurston–Bennequin number, and 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. 969–995DOI 10.4171/JEMS/383