Cables of thin knots and bordered Heegaard Floer homology

TL;DR

This paper develops formulas using bordered Floer homology to compute knot Floer homology and concordance invariants for cable knots, especially thin knots, in terms of known invariants and parameters.

Contribution

It provides explicit formulas for the knot Floer homology and tau invariants of cable knots based on the properties of the original thin knot, expanding computational tools in knot theory.

Findings

Formulas for knot Floer homology of (p, pn+1)-cables of thin knots.

Explicit expression for tau invariants of cable knots.

Applicable to almost all relatively prime p and q for tau(K_{p,q}).

Abstract

We use bordered Floer homology to give a formula for the knot Floer homology of any (p, pn+1)-cable of a thin knot K in terms of Delta_K(t), tau(K), p, and n. We also give a formula for the Ozsvath-Szabo concordance invariant tau(K_{p, pn+1}) in terms of tau(K), p, and n, and a formula for tau(K_{p,q}) for almost all relatively prime p and q.