Combinatorial knot Floer homology and cyclic branched covers

TL;DR

This paper provides a combinatorial proof of the invariance of knot Floer homology in cyclic branched covers using Heegaard diagrams derived from grid diagrams, advancing computational methods in knot theory.

Contribution

It introduces a combinatorial approach to prove invariance of knot Floer homology in cyclic branched covers, utilizing Heegaard diagrams from grid diagrams.

Findings

Proves invariance of combinatorial knot Floer homology over integers.

Develops a method to construct Heegaard diagrams from grid diagrams.

Provides a combinatorial framework for studying knot Floer homology in branched covers.

Abstract

Using a Heegaard diagram for the pullback of a knot $K \subset S^3$ in its cyclic branched cover $\Sigma_m(K)$ obtained from a grid diagram for $K$, we give a combinatorial proof for the invariance of the associated combinatorial knot Floer homology over $\mathbb{Z}$.