On the knot Floer homology of a class of satellite knots

TL;DR

This paper investigates the behavior of knot Floer homology for a specific class of satellite knots, revealing how the invariant relates to the Alexander-Conway polynomials of the satellite, companion, and pattern, and exploring implications for the Seifert genus.

Contribution

It generalizes the relationship between Alexander-Conway polynomials and knot Floer homology for satellite knots and examines the Seifert genus in this context.

Findings

Established a relation between knot Floer homology and Alexander-Conway polynomials for satellite knots.

Provided insights into how the Seifert genus can be studied via Floer homology.

Extended classical invariants to a broader class of knots.

Abstract

Knot Floer homology is an invariant for knots in the three-sphere for which the Euler characteristic is the Alexander-Conway polynomial of the knot. The aim of this paper is to study this homology for a class of satellite knots, so as to see how a certain relation between the Alexander-Conway polynomials of the satellite, companion and pattern is generalized on the level of the knot Floer homology. We also use our observations to study a classical geometric invariant, the Seifert genus, of our satellite knots.