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

TL;DR

This paper establishes a genus bound for knots in S^3 based on p-surgery results that produce L-spaces, and confirms the cabling conjecture for surgeries yielding connected sums of lens spaces.

Contribution

It proves a new genus bound related to p-surgery producing L-spaces and confirms the cabling conjecture for certain lens space surgeries.

Findings

Established a genus bound: 2g(K) - 1 q p - \u221a{3p+1}

Constructed infinite families of knots attaining the bound

Confirmed the cabling conjecture for surgeries resulting in lens space sums

Abstract

We prove that if positive integer p-surgery along a knot K \subset S^3 produces an L-space and it bounds a sharp 4-manifold, then the knot genus obeys the bound 2g(K) -1 \leq p - \sqrt{3p+1}. Moreover, there exists an infinite family of pairs (K_n,p_n) attaining this bound, where K_n denotes an n-fold iterated cable of the unknot and p_n \to \infty. In particular, the stated bound applies when the knot surgery produces 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.