A note on knot Floer homology and fixed points of monodromy

TL;DR

This paper proves that for certain hyperbolic fibered knots with specific knot Floer homology properties, the monodromy map is isotopic to a pseudo-Anosov map without fixed points, revealing new insights into knot dynamics.

Contribution

It establishes a link between the rank of a specific knot Floer homology group and the fixed point properties of the monodromy map for hyperbolic fibered knots.

Findings

Monodromy of certain hyperbolic fibered knots is fixed point free.

Hyperbolic L-space knots have monodromy isotopic to fixed point free maps.

Knot Floer homology rank constrains monodromy dynamics.

Abstract

Using an argument of Baldwin--Hu--Sivek, we prove that if $K$ is a hyperbolic fibered knot with fiber $F$ in a closed, oriented $3$--manifold $Y$, and $\widehat{HFK}(Y,K,[F], g(F)-1)$ has rank $1$, then the monodromy of $K$ is freely isotopic to a pseudo-Anosov map with no fixed points. In particular, this shows that the monodromy of a hyperbolic L-space knot is freely isotopic to a map with no fixed points.