Cables of thin knots and bordered Heegaard Floer homology

Abstract

We use bordered Floer homology to give a formula for $\widehat{\mathrm{HFK}}(K_{p,pn+1})$ 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 Ozsváth–Szabó concordance invariant $\tau (K_{p,q})$ in terms of $\tau (K)$, $p$, and $q$, for all relatively prime $p$ and $q$.