Symmetries and adjunction inequalities for knot Floer homology

TL;DR

This paper explores symmetries and inequalities in knot Floer homology, providing new tools for understanding knot invariants, especially for homologically essential knots, and demonstrating their applications through explicit calculations.

Contribution

It introduces new symmetry properties and adjunction inequalities for knot Floer homology, including for cobordism maps, and applies these to distinguish fibered knots and recover key results.

Findings

Derived symmetries and adjunction inequalities for knot Floer homology.

Established vanishing results and explicit calculations illustrating the inequalities.

Proved that knot Floer homology distinguishes certain fibered knots from others.

Abstract

We derive symmetries and adjunction inequalities of the knot Floer homology groups which appear to be especially interesting for homologically essential knots. Furthermore, we obtain an adjunction inequality for cobordism maps in knot Floer homologies. We demonstrate the adjunction inequalities and symmetries in explicit calculations which recover some of the main results from [1] on longitude Floer homology and also give rise to vanishing results on knot Floer homologies. Furthermore, using symmetries we prove that the knot Floer homology of a fiber distinguishes $\stwo\times\sone$ from other $\sone$-bundles over surfaces.