Non-contractible loops of Legendrian tori from families of knots

TL;DR

This paper introduces a topological method using knot cord algebra to compute monodromy on Legendrian contact homology, revealing infinite non-contractible loops of Legendrian tori in the unit cotangent bundle of 3.

Contribution

It presents a novel topological approach to compute monodromy on Legendrian contact homology using knot invariants, identifying new non-contractible loops of Legendrian tori.

Findings

Infinite family of non-contractible Legendrian tori loops

Method to compute monodromy via cord algebra

Loops are contractible in smooth embedding space

Abstract

In the unit cotangent bundle of $\mathbb{R}^3$, we consider loops of Legendrian tori arising as families of the unit conormal bundles of smooth knots in $\mathbb{R}^3$. In this paper, using the cord algebra of knots, we give a topological method to compute the monodromy on the Legendrian contact homology in degree $0$ induced by those loops. As an application, we obtain an infinite family of non-contractible loops of Legendrian tori which are contractible in the space of smoothly embedded tori.