Knotted solid tori in contact manifolds

TL;DR

This paper analyzes solid tori in contact manifolds to understand when knot widths match the maximal Thurston–Bennequin invariant and when non-thickenable tori arise. By combining convex surface theory, Farey-graph classifications, and splitting/symplectic-filling techniques, it proves $w(K)=\overline{\mathrm{tb}}(K)$ for broad classes of knots, notably $L$-space knots with $\overline{\mathrm{tb}}(K)=2g(K)-1$, and establishes width-one behavior for many cables under suitable slope conditions. It also demonstrates that the ambient contact structure on $S^3$ restricted to a solid torus is universally tight except in cases involving Lagrangian-slice knots with slopes in $(-\tfrac{1}{2},0)$, and it constructs extensive families of non-thickenable tori in various knot types beyond $S^3$, including genus-1 open books and those with trivial monodromy. The results yield new criteria for uniform thickness, illuminate the role of Legendrian large cables, and open pathways toward a broader conjectural picture connecting width, tb, and the contact-topological support of knots.

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 $S^3$.

Theorems & Definitions (27)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• proof
• Theorem 1.4
• Conjecture 1.5
• Theorem 1.6
• Conjecture 1.7
• Conjecture 1.8
• Theorem 1.9