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

Cagatay Kutluhan, Gordana Matic, Jeremy Van Horn-Morris, Andy Wand

TL;DR

This work addresses computing Heegaard Floer invariants for closed contact 3-manifolds from open books by introducing two SageMath programs, hf-hat-obd and hf-hat-obd-nice, plus an auxiliary makenice. It proves an a priori bound on the number of positive domains between generators and develops a practical framework to detect finiteness of the spectral order, enabling computation of the contact invariant and spectral order via open-book data. The paper provides concrete tutorials for both tools, including how to obtain nice Heegaard diagrams and how to extract the relevant invariants. Collectively, the methods advance a computational pipeline for verifying Stein fillability obstructions and related contact-geometric properties through explicit Heegaard Floer calculations.

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 Ozsváth--Szabó 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 Ozsváth--Szabó contact invariant.

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

TL;DR

This work addresses computing Heegaard Floer invariants for closed contact 3-manifolds from open books by introducing two SageMath programs, hf-hat-obd and hf-hat-obd-nice, plus an auxiliary makenice. It proves an a priori bound on the number of positive domains between generators and develops a practical framework to detect finiteness of the spectral order, enabling computation of the contact invariant and spectral order via open-book data. The paper provides concrete tutorials for both tools, including how to obtain nice Heegaard diagrams and how to extract the relevant invariants. Collectively, the methods advance a computational pipeline for verifying Stein fillability obstructions and related contact-geometric properties through explicit Heegaard Floer calculations.

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 Ozsváth--Szabó 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 Ozsváth--Szabó contact invariant.

Paper Structure

This paper contains 6 sections, 6 theorems, 18 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main Results
  3. A bound on the number of positive domains
  4. Detecting finiteness of spectral order
  5. A Tutorial for hf-hat-obd
  6. A tutorial for hf-hat-obd-nice

Key Result

Theorem 1

KMVHMW1 If the contact structure is Stein fillable, then its spectral order is infinite.

Figures (2)

  • Figure 1: Neighborhood of a contact intersection point.
  • Figure 2: Directed graph depicting $b_0,\dots,b_k\in C_1$ from \ref{['eq:zigzag']} and their differentials.

Theorems & Definitions (11)

  • Theorem
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Theorem 5
  • ...and 1 more