Legendrian and transverse twist knots

By John B. Etnyre, Lenhard L. Ng, Vera Vértesi. 2010

60 citations recorded by the source corpus.

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.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/95332b2c4ef12f59e0819762aa1710279430c0ed

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

Citations in the corpus

Infinitely many Lagrangian fillings (2020, background): “Legendrian knots in contact 3–manifolds are instrumental to study the contact geometry of 3–manifolds [6, 23, 24, 25, 27, 28, 37].”
Grid Homology for Knots and Links (2015, background): “The Legendrian knots in the knot type under examination are classified in [48]: in particular, there are exactly two distinct Legendrian knots in the knot type m(52) with tb = 1 and r = 0.”