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

TL;DR

This paper introduces a combinatorial method using Lagrangian hypercube diagrams to compute the rotation class and Maslov index of Legendrian tori in 5^5, aiding contact homology calculations.

Contribution

It provides a new combinatorial approach to compute the rotation class and Maslov index of Legendrian tori, which was previously challenging.

Findings

Derived a formula for the Maslov index of loops on Legendrian tori.

Computed the Maslov number for a broad family of Legendrian tori.

Established the use of Lagrangian hypercube diagrams for these computations.

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.