Legendrian and transverse twist knots

TL;DR

This paper provides a complete classification of Legendrian and transverse representatives of twist knots, revealing exact counts of maximal Thurston--Bennequin and self-linking number representatives using advanced topological techniques.

Contribution

It introduces a full classification of twist knot representatives, quantifying their Legendrian and transverse forms with novel counts and methods.

Findings

$K_{-2n}$ has $ ceilrac{n^2}{2} ceil$ Legendrian representatives with maximal Thurston--Bennequin number.

$K_{-2n}$ has $ ceilrac{n}{2} ceil$ transverse representatives with maximal self-linking number.

Utilizes convex surface theory, Legendrian ruling invariants, and Heegaard Floer homology.

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.