To compute orientations of Morse flow trees in Legendrian contact homology

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 $n=1$.