Knotted solid tori in contact manifolds

Abstract

In this paper, we study solid tori in contact manifolds. Specifically, we study the contact width of a knot type and give criteria for when it can be explicitly computed. We also prove there are many “nonthickenable” tori in many knot types. These tori are frequently essential in the study of Legendrian and transverse knot theory and tight contact structures on manifolds obtained by surgery on the knot. Previously, nonthickenable tori have only been observed for iterated torus knots in S 3 and were thought to be rare. We show that they are quite common, exist in other manifolds, and even for a hyperbolic knot in S 3 . We also make and highlight several conjectures about the general nature of knots in contact manifolds.