A note on cabling and L-space surgeries

TL;DR

This paper characterizes when the (p,q)-cable of a knot in S^3 admits a positive L-space surgery, linking it to the original knot's L-space surgeries and the ratio q/p relative to the knot's genus.

Contribution

It provides a complete if-and-only-if condition for positive L-space surgeries on cable knots based on the original knot's properties and the cabling parameters.

Findings

The (p,q)-cable of a knot admits a positive L-space surgery if and only if the original knot does and q/p   2g(K)-1.

The result extends Hedden's work by establishing the necessary and sufficient condition.

The condition involves the Seifert genus of the original knot and the cabling ratio q/p.

Abstract

We prove that the (p,q)-cable of a knot K in S^3 admits a positive L-space surgery if and only if K admits a positive L-space surgery and q/p \geq 2g(K)-1, where g(K) is the Seifert genus of K. The "if" direction is due to Hedden.