Knotted solid tori in contact manifolds

TL;DR

This paper investigates the properties of solid tori within contact manifolds, focusing on knot width, Thurston-Bennequin invariants, and the existence of non-thickenable tori across various knot types.

Contribution

It provides new criteria for knot width relative to Thurston-Bennequin invariants and demonstrates the existence of non-thickenable tori in diverse knot types beyond $S^3$.

Findings

Criteria for when knot width equals the maximal Thurston-Bennequin invariant.

Identification of conditions where knot width exceeds the maximal Thurston-Bennequin invariant.

Existence of many non-thickenable tori in various knot types.

Abstract

In this note we study solid tori in contact manifolds. Specifically, we study the width of a knot type and give criteria for when it is equal to the maximal Thurston-Bennequin invariant, and when it is larger. We also prove there are many ``non-thickenable" tori in many knot types. These had previously only been observed in $S^3$.