Legendrian Torus and Cable Links

TL;DR

This paper classifies Legendrian torus and cable links, revealing new phenomena such as links with symmetries not realizable by Legendrian isotopies and non-destabilizable links lacking maximal Thurston-Bennequin invariant.

Contribution

It provides the first classification of infinite families of Legendrian links with certain symmetry properties and extends the classification to Legendrian and transversal cable links of specific knot types.

Findings

Identified Legendrian links with non-realizable symmetries.

Discovered non-destabilizable links without maximal Thurston-Bennequin invariant.

Classified Legendrian and transversal cable links of uniformly thick, Legendrian simple knots.

Abstract

We give a classification of Legendrian torus links. Along the way, we give the first classification of infinite families of Legendrian links where some smooth symmetries of the link cannot be realized by Legendrian isotopies. We also give the first family of links that are non-destabilizable but do not have maximal Thurston-Bennequin invariant and observe a curious distribution of Legendrian torus knots that can be realized as the components of a Legendrian torus link. This classification of Legendrian torus links leads to a classification of transversal torus links. We also give a classification of Legendrian and transversal cable links of knot types that are uniformly thick and Legendrian simple. Here we see some similarities with the classification of Legendrian torus links but also some differences. In particular, we show that there are Legendrian representatives of cable links of any uniformly thick knot type for which no symmetries of the components can be realized by a Legendrian isotopy, others where only cyclic permutations of the components can be realized, and yet others where all smooth symmetries are realizable.