Legendrian and transverse cables of positive torus knots

TL;DR

This paper classifies Legendrian and transverse knots in cable types of positive torus knots, revealing new phenomena such as non-destabilizable knots with arbitrarily low Thurston-Bennequin invariants and complex stabilization behaviors.

Contribution

It introduces the concept of partially thickenable tori to fully classify solid tori representing positive torus knots, enabling new insights into knot theory.

Findings

Existence of non-destabilizable Legendrian knots with arbitrarily low Thurston-Bennequin invariant

Legendrian knots requiring many stabilizations before isotopy

New phenomena observed for transverse knots

Abstract

In this paper we classify Legendrian and transverse knots in the knot types obtained from positive torus knots by cabling. This classification allows us to demonstrate several new phenomena. Specifically, we show there are knot types that have non-destabilizable Legendrian representatives whose Thurston-Bennequin invariant is arbitrarily far from maximal. We also exhibit Legendrian knots requiring arbitrarily many stabilizations before they become Legendrian isotopic. Similar new phenomena are observed for transverse knots. To achieve these results we define and study "partially thickenable" tori, which allow us to completely classify solid tori representing positive torus knots.