Simplified SFT moduli spaces for Legendrian links

Abstract

We study moduli spaces $\mathcal{M}$ of holomorphic maps $U$ from Riemann surfaces to ${\mathbb{R}}^4$ with boundaries on the Lagrangian cylinder over a Legendrian link $\Lambda \subset ({\mathbb{R}}^3,{\xi }_{std})$. We allow our domains, $\Sigma$, to have non-trivial topology in which case $\mathcal{M}$ is the zero locus of an obstruction function $\mathcal{O}$, sending a moduli space of holomorphic maps in $\mathbb{C}$ to $H^1(\Sigma )$. In general, ${\mathcal{O}}^{-1}(0)$ is not combinatorially computable. However after a Legendrian isotopy, $\Lambda$ can be made left-right-simple, implying that any $U$ of index $1$ is a disk with one or two positive punctures for which ${\pi }_{\mathbb{C}}\circ U$ is an embedding. Moreover, any $U$ of index $2$ is either a disk or an annulus with ${\pi }_{\mathbb{C}}\circ U$ simply covered and without interior critical points. Therefore any SFT invariant of $\Lambda$ is combinatorially computable using only disks with $\le 2$ positive punctures.

42 pages. V4: Updates based on referee feedback

References in corpus (2)

• Semi-global Kuranishi charts and the definition of contact homology
• Combinatorial Reeb dynamics on punctured contact 3-manifolds