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

TL;DR

This paper introduces SageMath programs for computing Heegaard Floer invariants of closed contact 3-manifolds from open books, enhancing analysis and detection of Stein fillability obstructions.

Contribution

It presents new software tools that automate and improve the computation of Heegaard Floer invariants from open books, including generating nice diagrams and analyzing contact invariants.

Findings

Programs successfully compute Heegaard Floer invariants

Tools detect finiteness of spectral order as a Stein fillability obstruction

Enhanced analysis of contact structures via computational methods

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.