Computing Heegaard Floer invariants of closed contact 3-manifolds from open books

Abstract

We present two SageMath programs that build on and improve upon Sucharit Sarkar's hf-hat. Given an abstract open book and a collection of pairwise disjoint properly embedded arcs on a page of the open book, the first program, hf-hat-obd, can be used to analyze the resulting Heegaard diagram, while the second, hf-hat-obd-nice computes the hat version of Heegaard Floer homology of the closed oriented 3-manifold described by the Heegaard diagram as long as the latter is nice. We also provide an auxiliary program, makenice, that can be used to produce a nice Heegaard diagram out of any abstract open book and a collection of pairwise disjoint properly embedded arcs on a page of the open book. The primary applications of hf-hat-obd-nice are to the computation of the Ozsv\'ath--Szab\'o contact invariant and to the detection of finiteness of spectral order, which is a Stein fillability obstruction that is stronger than the vanishing of the Ozsv\'ath--Szab\'o contact invariant.