Cable links and L-space surgeries

TL;DR

This paper characterizes when cable links are L-space links based on properties of the original knot and the ratio of parameters, and computes their link Floer homology, confirming a conjecture for torus links.

Contribution

It provides a complete criterion for L-space cable links and explicit computations of their link Floer homology, advancing understanding of L-space links.

Findings

Cable link $K_{rm,rn}$ is an L-space link iff K is an L-space knot and $n/m \\geq 2g(K)-1$

Explicit formulas for HFL-minus and HFL-hat of L-space cable links

Confirmed Licata's conjecture on HFL-hat structure for (n,n) torus links

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 $n/m \geq 2g(K)-1$. We also compute HFL-minus and HFL-hat of an L-space cable link in terms of its Alexander polynomial. As an application, we confirm a conjecture of Licata regarding the structure of HFL-hat for (n,n) torus links.