A combinatorial description of the LOSS Legendrian knot invariant

TL;DR

This paper demonstrates that the hat version of the Heegaard Floer invariant for Legendrian knots in contact three-manifolds can be computed combinatorially, leveraging Plamenevskaya's description, simplifying calculations in contact topology.

Contribution

It provides a combinatorial method to compute the Heegaard Floer Legendrian knot invariant, making the process more accessible and computationally feasible.

Findings

The invariant can be computed combinatorially.

The approach simplifies calculations in contact topology.

It connects Heegaard Floer invariants with combinatorial descriptions.

Abstract

In this note, we observe that the hat version of the Heegaard Floer invariant of Legendrian knots in contact three-manifolds defined by Lisca-Ozsvath-Stipsicz-Szabo can be combinatorially computed. We rely on Plamenevskaya's combinatorial description of the Heegaard Floer contact invariant.