Exceptional Legendrian torus knots

TL;DR

This paper classifies exceptional Legendrian realizations of torus knots, providing the first such results for non-trivial topological knots and establishing bounds on tight contact structures.

Contribution

It introduces the first classification of exceptional Legendrian torus knots and uses contact surgery diagrams to demonstrate realizations, advancing understanding of contact topology.

Findings

Bounds on tight contact structures on knot complements

Classification of exceptional Legendrian torus knots

Existence theorems for exceptional Legendrian knots

Abstract

We present classification results for exceptional Legendrian realisations of torus knots. These are the first results of that kind for non-trivial topological knot types. Enumeration results of Ding-Li-Zhang concerning tight contact structures on certain Seifert fibred manifolds with boundary allow us to place upper bounds on the number of tight contact structures on the complements of torus knots; the classification of exceptional realisations of these torus knots is then achieved by exhibiting sufficiently many realisations in terms of contact surgery diagrams. We also discuss a couple of general theorems about the existence of exceptional Legendrian knots.