Non-contractible loops of Legendrian tori from families of knots

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.