Cable links and L-space surgeries

Abstract

An L-space link is a link in $S^3$ on which all sufficiently large integral surgeries are L-spaces. We prove that for $m,n$ relatively prime, the $r$-component cable link $K_{rm,rn}$ is an L-space link if and only if $K$ is an L-space knot and $n/m\ge 2g(K)-1$. We also compute ${\mathrm{HFL}}^{–}$ and $\widehat{\mathrm{HFL}}$ of an L-space cable link in terms of its Alexander polynomial. As an application, we confirm a conjecture of Licata [7] regarding the structure of $\widehat{\mathrm{HFL}}$ for $(n,n)$ torus links.