A Combinatorial Description of the Knot Concordance Invariant Epsilon

TL;DR

This paper provides a combinatorial framework for the knot concordance invariant epsilon, computes it for specific knots, and explores its behavior under cabling, enhancing understanding of knot concordance properties.

Contribution

It introduces a combinatorial description of epsilon, proves its properties with grid homology, and analyzes its behavior under cabling operations.

Findings

Computed epsilon for (p,q) torus knots.

Proved epsilon equals 1 for positive braid grid diagrams.

Analyzed epsilon's behavior under (p,q)-cabling of negative torus knots.

Abstract

In this paper, we give a combinatorial description of the concordance invariant $\varepsilon$ defined by Hom in \cite{hom2011knot}, prove some properties of this invariant using grid homology techniques. We also compute $\varepsilon$ of $(p,q)$ torus knots and prove that $\varepsilon(\mathbb{G}_+)=1$ if $\mathbb{G}_+$ is a grid diagram for a positive braid. Furthermore, we show how $\varepsilon$ behaves under $(p,q)$-cabling of negative torus knots.