Maximal Thurston-Bennequin number and reducible Legendrian surgery

TL;DR

This paper introduces a method using Stein handle decompositions to construct Legendrian knots with maximal Thurston-Bennequin numbers and demonstrates that infinitely many knots can produce reducible 3-manifolds via Legendrian surgery, disproving a conjecture.

Contribution

It provides a new construction technique for Legendrian knots and shows the existence of infinitely many knots with reducible Legendrian surgeries, countering previous conjectures.

Findings

Constructed Legendrian representatives with maximal Thurston-Bennequin number.

Produced infinitely many knots yielding reducible 3-manifolds via Legendrian surgery.

Disproved a conjecture of Lidman and Sivek.

Abstract

We give a method for constructing a Legendrian representative of a knot in $S^3$ which realizes its maximal Thurston-Bennequin number under a certain condition. The method utilizes Stein handle decompositions of $D^4$, and the resulting Legendrian representative is often very complicated (relative to the complexity of the topological knot type). As an application, we construct infinitely many knots in $S^3$ each of which yields a reducible 3-manifold by a Legendrian surgery in the standard tight contact structure. This disproves a conjecture of Lidman and Sivek.