On the rotation class of knotted Legendrian Tori in $\mathbb{R}^5$

Abstract

In this paper we show how to combinatorically compute the rotation class of a large family of embedded Legendrian tori in $\mathbb{R}^5$ with the standard contact form. In particular, we give a formula to compute the Maslov index for any loop on the torus and compute the Maslov number of the Legendrian torus. These formulas are a necessary component in computing contact homology. Our methods use a new way to represent knotted Legendrian tori called Lagrangian hypercube diagrams.