L-space satellite operators and knot Floer homology

TL;DR

This paper develops a formula to compute the knot Floer complex of satellite knots with L-space link companions, using Heegaard Floer techniques and properties of L-space links, with practical implementation in Python.

Contribution

It introduces a new satellite formula for knot Floer complexes involving L-space links and proves that their link Floer complexes are determined by multivariable Alexander polynomials.

Findings

Derived a formula for satellite knot Floer complexes using Heegaard Floer techniques.

Proved that 2-component L-space links have formal knot Floer complexes.

Implemented the satellite formula in Python for practical computations.

Abstract

We consider satellite operators where the corresponding 2-component link is an L-space link. This family includes many commonly studied satellite operators, including cabling operators, the Whitehead operator, and a family of Mazur operators. We give a formula which computes the knot Floer complex of a satellite of $K$ in terms of the knot Floer complex of $K$. Our main tools are the Heegaard Floer Dehn surgery formulas and their refinements. A key step in our computation is a proof that 2-component L-space links have formal knot Floer complexes. We use this to show that the link Floer complexes of 2-component L-space links are determined by their multivariable Alexander polynomials. We implement our satellite formula in Python code, which we also make available.