Heegaard Floer homology and concordance bounds on the Thurston norm

TL;DR

This paper uses Heegaard Floer homology to establish new lower bounds on the Thurston norm and geometric winding number for certain links and knots, providing obstructions and sharp examples.

Contribution

It introduces twisted correction terms in Heegaard Floer homology as tools for bounding the Thurston norm and winding number, and constructs explicit knots with sharp bounds.

Findings

Lower bounds on Thurston norm via twisted correction terms

Obstructions for knots to have untwisting number 1

Infinite family of knots with sharp bounds on winding number

Abstract

We prove that twisted correction terms in Heegaard Floer homology provide lower bounds on the Thurston norm of certain cohomology classes determined by the strong concordance class of a 2-component link $L$ in $S^3$. We then specialise this procedure to knots in $S^2\times S^1$, and obtain a lower bound on their geometric winding number. Furthermore we produce an obstruction for a knot in $S^3$ to have untwisting number 1. We then provide an infinite family of null-homologous knots with increasing geometric winding number, on which the bound is sharp.