Berge–Gabai knots and L–space satellite operations

Abstract

Let $P\left(K\right)$ be a satellite knot where the pattern $P$ is a Berge–Gabai knot (ie a knot in the solid torus with a nontrivial solid torus Dehn surgery) and the companion $K$ is a nontrivial knot in $S^3$. We prove that $P\left(K\right)$ is an L–space knot if and only if $K$ is an L–space knot and $P$ is sufficiently positively twisted relative to the genus of $K$. This generalizes the result for cables due to Hedden [Int. Math. Res. Not. 2009 (2009) 2248–2274] and Hom [Algebr. Geom. Topol. 11 (2011) 219–223].