Bordered Heegaard Floer homology and the tau-invariant of cable knots

TL;DR

This paper introduces a new concordance invariant epsilon(K) from knot Floer homology and provides a formula to compute the tau-invariant for cable knots, enhancing understanding of knot concordance and cabling effects.

Contribution

It defines the epsilon invariant and derives a formula for tau of cable knots based on epsilon, tau, p, and q, advancing knot concordance theory.

Findings

Epsilon(K) is a new concordance invariant derived from knot Floer complex.

A formula for tau of (p,q)-cable knots in terms of epsilon, tau, p, and q is established.

Properties of epsilon under cabling are characterized, enabling computation of tau for iterated cables.

Abstract

We define a concordance invariant, epsilon(K), associated to the knot Floer complex of K, and give a formula for the Ozsv\'ath-Szab\'o concordance invariant tau of K_{p,q}, the (p,q)-cable of a knot K, in terms of p, q, tau(K), and epsilon(K). We also describe the behavior of epsilon under cabling, allowing one to compute tau of iterated cables. Various properties and applications of epsilon are also discussed.