To compute orientations of Morse flow trees in Legendrian contact homology

TL;DR

This paper presents an algorithm to compute orientations of Morse flow trees in Legendrian contact homology, enabling integer coefficient calculations for Legendrian invariants in higher dimensions.

Contribution

It introduces a novel algorithm for orienting moduli spaces of Morse flow trees in Legendrian contact homology, extending computations beyond the case of dimension one.

Findings

Algorithm for computing orientations of Morse flow trees

Orientation determination up to 4 signs in dimension one

Facilitates integer coefficient Legendrian contact homology calculations

Abstract

Let $\Lambda$ be a closed, connected Legendrian submanifold of the 1-jet space of a smooth $n$-dimensional manifold. Associated to $\Lambda$ there is a Legendrian invariant called Legendrian contact homology, which is defined by counting rigid pseudo-holomorphic disks of $\Lambda$. Moreover, there exists a bijective correspondence between rigid pseudo-holomorphic disks and rigid Morse flow trees of $\Lambda$, which allows us to compute the Legendrian contact homology of $\Lambda$ via Morse theory. If $\Lambda$ is spin, then the moduli space of the rigid disks can be given a coherent orientation, so that the Legendrian contact homology of $\Lambda$ can be defined with coefficients in $\mathbb Z$. In this paper we give an algorithm for computing the corresponding orientation of the moduli space of rigid Morse flow trees if the dimension of $\Lambda $ is greater than 1, and up to 4 signs that depend on data that can be extracted from the vertices of the trees, but which are not given explicitly, in the case $n=1$.