Computing Knot Floer Homology in Cyclic Branched Covers

TL;DR

This paper introduces a combinatorial algorithm using grid diagrams to compute knot Floer homology in cyclic branched covers, providing explicit calculations for numerous knots with up to eleven crossings.

Contribution

It presents a novel combinatorial method for calculating knot Floer homology in cyclic branched covers, expanding computational tools in knot theory.

Findings

Computed knot Floer homology for over fifty three-bridge knots with up to eleven crossings in the double cover.

Developed a new algorithm based on grid diagrams for these calculations.

Demonstrated the effectiveness of the method through extensive examples.

Abstract

We use grid diagrams to give a combinatorial algorithm for computing the knot Floer homology of the pullback of a knot K in its m-fold cyclic branched cover Sigma^m(K), and we give computations when m=2 for over fifty three-bridge knots with up to eleven crossings.