On knot Floer homology for some fibered knots

TL;DR

This paper reestablishes key results in knot Floer homology for fibered knots, emphasizing integer coefficients and braid structures, and applies these to derive properties of associated three-manifolds.

Contribution

It provides a new proof of knot Floer homology results for alternating branched loci without Khovanov homology, enhancing applicability and clarity.

Findings

Results hold for integer coefficients

Braid structures reveal knot Floer homology information

Establishes parallels with Heegaard-Floer results for positive braids

Abstract

A companion paper to "On knot Floer homology in branched double covers" applied to braided branched loci. We reprove the main result of that paper concerning alternating branched loci when projected to an annulus, without using Khovanov homology. This provides two advantages: 1) the results hold for integer coefficients and 2) the spin^c structures are more readily discernable. We apply this result to a branch locus which is a braid, and use the braid structure to find information about a fibered knot in the branched double cover. In some cases this provides all the information about the knot Floer homology and can be used to derive information about the Heegaard-Floer homology of associated fibered three manifolds. Results for certain positive braids are also included, establishing results similar to E. Eftekhary's in the Heegaard-Floer setting.