Heegaard Floer homology and maximal twisting numbers

TL;DR

This paper classifies negative-twisting tight contact structures on Seifert fibred spaces over S^2 using Heegaard Floer homology, establishing a complete correspondence and explicit invariants.

Contribution

It adapts the Ozsváth-Szabó algorithm to star-shaped graphs and provides a full classification of negative-twisting structures on these manifolds.

Findings

Complete classification of negative-twisting structures on Seifert fibred spaces.

Structures are distinguished by their contact invariant c^+.

Number of structures is combinatorially determined by Seifert coefficients.

Abstract

We adapt the Ozsv\'ath-Szab\'o full path algorithm to every star-shaped graph and establish a correspondence between negative-twisting tight contact structures on any Seifert fibred space over $S^2$, and its Heegaard Floer homology groups equipped with the Alexander filtration induced by the regular fibre. This provides the complete classification of negative-twisting structures on these manifolds; in particular, we distinguish them by their contact invariant $c^+$. We prove that every such structure is symplectically fillable and extend a known obstruction to Stein fillability. In addition, we show that the number of negative-twisting structures can be expressed combinatorially in terms of the Seifert coefficients of the star-shaped graph, while their $d_3$-invariant and homotopy type are determined explicitly through our correspondence. Our results also complete the classification of fillable structures on any small Seifert fibred space.