L-space surgeries, genus bounds, and the cabling conjecture

Abstract

We establish a tight inequality relating the knot genus $g(K)$ and the surgery slope $p$ under the assumption that $p$-framed Dehn surgery along $K$ is an L-space that bounds a sharp $4$-manifold. This inequality applies in particular when the surgered manifold is a lens space or a connected sum thereof. Combined with work of Gordon–Luecke, Hoffman, and Matignon–Sayari, it follows that if surgery along a knot produces a connected sum of lens spaces, then the knot is either a torus knot or a cable thereof, confirming the cabling conjecture in this case.