Floer Homology and Fractional Dehn Twists

TL;DR

This paper links Heegaard Floer homology to fractional Dehn twist coefficients, providing bounds on contact structures and braid representations, advancing understanding of 3-manifold and knot invariants.

Contribution

It establishes a new relationship between Floer homology ranks and fractional Dehn twist coefficients, connecting contact topology and braid theory.

Findings

Floer homology bounds the fractional Dehn twist coefficient of surface automorphisms.

Rank of Floer homology constrains the number of boundary twists for tight or overtwisted contact structures.

Khovanov homology rank bounds the fractional Dehn twist coefficient of braid representatives.

Abstract

We establish a relationship between Heegaard Floer homology and the fractional Dehn twist coefficient of surface automorphisms. Specifically, we show that the rank of the Heegaard Floer homology of a 3-manifold bounds the absolute value of the fractional Dehn twist coefficient of the monodromy of any of its open book decompositions with connected binding. We prove this by showing that the rank of Floer homology gives bounds for the number of boundary parallel right or left Dehn twists necessary to add to a surface automorphism to guarantee that the associated contact manifold is tight or overtwisted, respectively. By examining branched double covers, we also show that the rank of the Khovanov homology of a link bounds the fractional Dehn twist coefficient of its odd-stranded braid representatives.