A note on cabling and $L$–space surgeries

Abstract

We prove that the $\left(p,q\right)$–cable of a knot $K\subset S^3$ admits a positive $L$–space surgery if and only if $K$ admits a positive $L$–space surgery and $q∕p\ge 2g\left(K\right)-1$, where $g\left(K\right)$ is the Seifert genus of $K$. The “if” direction is due to Hedden [Int. Math. Res. Not. 2009 (2009) 2248–2274].