Contact structures and reducible surgeries

TL;DR

This paper combines contact topology and surgery theory to determine when Legendrian surgery produces reducible manifolds, proving the cabling conjecture for certain positive knots and establishing bounds on Thurston-Bennequin numbers.

Contribution

It introduces new techniques linking contact topology with surgery theory to classify reducible surgeries on positive knots and proves the cabling conjecture for genus 2 positive knots.

Findings

Reducible surgery on non-cabled positive knots of genus g occurs at slope 2g-1.

Proved the cabling conjecture for positive knots of genus 2.

Derived bounds on maximum Thurston-Bennequin numbers of cables.

Abstract

We apply results from both contact topology and exceptional surgery theory to study when Legendrian surgery on a knot yields a reducible manifold. As an application, we show that a reducible surgery on a non-cabled positive knot of genus g must have slope 2g-1, leading to a proof of the cabling conjecture for positive knots of genus 2. Our techniques also produce bounds on the maximum Thurston-Bennequin numbers of cables.