Heegaard Floer homology and genus one, one boundary component open books

TL;DR

This paper computes Heegaard Floer homology for certain 3-manifolds with genus one open books and torus bundles, providing insights into contact structures, knot concordance, and quasi-alternating links.

Contribution

It offers explicit Heegaard Floer homology calculations for genus one open books and related manifolds, linking these to contact geometry and knot theory.

Findings

Restrictions on Stein-fillings of compatible contact structures

Characterization of 3-braid knots with finite concordance order

Identification of all quasi-alternating links with braid index ≤ 3

Abstract

We compute the Heegaard Floer homology of any rational homology 3-sphere with an open book decomposition of the form (T,\phi), where T is a genus one surface with one boundary component. In addition, we compute the Heegaard Floer homology of any T^2-bundle over S^1 with first Betti number equal to one, and we compare our results with those of Lebow on the embedded contact homology of such torus bundles. We use these computations to place restrictions on Stein-filllings of the contact structures compatible with such open books, to narrow down somewhat the class of 3-braid knots with finite concordance order, and to identify all quasi-alternating links with braid index at most 3.