Knot Floer homology and fixed points

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.

13 pages. V2: We incorporate the referee's comments, and restructured Section 4 to include an overlooked case (Proposition 4.4). This is the version accepted for publication by the Journal of Topology