Knot Floer homology and fixed points

TL;DR

This paper links the rank of knot Floer homology to the minimal number of fixed points of the monodromy in fibered knots, extending previous results and clarifying a formula in symplectic Floer homology.

Contribution

It generalizes earlier work by establishing a bound on fixed points based on knot Floer homology rank and clarifies a computational formula in symplectic Floer homology.

Findings

Monodromy has at most r-1 fixed points when the Floer homology rank is r.

Generalizes Baldwin--Hu--Sivek and Ni's results.

Corrects a formula in Cotton-Clay's symplectic Floer homology computation.

Abstract

If $K$ is a fibered knot in a closed, oriented $3$--manifold $Y$ with fiber $F$, and $\widehat{HFK}(Y,K,[F], g(F)-1;\mathbb Z/2\mathbb Z)$ has rank $r$, then the monodromy of $K$ is freely isotopic to a diffeomorphism with at most $r-1$ fixed points. This generalizes earlier work of Baldwin--Hu--Sivek and Ni. We also clarify a misleading formula in Cotton-Clay's computation of the symplectic Floer homology of mapping classes of surfaces.