Rank Inequalities on Knot Floer Homology of Periodic Knots

TL;DR

This paper establishes a rank inequality relating the knot Floer homologies of a 2-periodic knot and its quotient, introduces a conjectured refinement, and explores applications to Alexander polynomials, advancing understanding in knot theory.

Contribution

It proves a new rank inequality for knot Floer homology of periodic knots and proposes a filtered refinement supported by computational evidence.

Findings

Proved a rank inequality between Floer homologies of periodic knots and their quotients.

Conjectured a refined inequality with computational support.

Applied results to relate Alexander polynomials of periodic knots.

Abstract

Let $\widetilde{K}$ be a 2-periodic knot in $S^3$ with quotient $K$. We prove a rank inequality between the knot Floer homology of $\widetilde{K}$ and the knot Floer homology of $K$ using a spectral sequence of Hendricks, Lipshitz and Sarkar. We also conjecture a filtered refinement of this inequality, for which we give computational evidence, and produce applications to the Alexander polynomials of $\widetilde{K}$ and $K$.