Cable links of uniformly thick knot types

TL;DR

This paper investigates Legendrian realizations of cable links in knot theory, revealing new phenomena and classification methods for complex cable links using Legendrian surgeries.

Contribution

It extends prior work by analyzing Legendrian cable links that are not Legendrian simple, introducing new techniques for classification based on Legendrian surgeries.

Findings

Discovery of stabilized Legendrian links that are smoothly isotopic but not Legendrian isotopic.

Classification results for Legendrian knots in negative cables of twist knots.

Development of a new technique using Legendrian surgeries for cable link analysis.

Abstract

In this paper, we study Legendrian realizations of cable links of knot types that are uniformly thick but not Legendrian simple, extending prior work of Dalton, the second author, and Traynor. This leads to new phenomena, such as stabilized Legendrian links that are smoothly isotopic and component-wise Legendrian isotopic, but are not Legendrian isotopic. In our study, we establish new results for cable links whose cabling slope is sufficiently negative. We will also show how to classify Legendrian knots in (most) negative cables of twist knots. This is done by introducing a new technique to the study of cables based on Legendrian surgeries.