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

2010/09/06 by Greene, Joshua Evan · 1 citation
#FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.1009.1130

Abstract

We prove that if positive integer p-surgery along a knot K ⊂ S³ produces an L-space and it bounds a sharp 4-manifold, then the knot genus obeys the bound 2g(K) -1 ≤ p - √(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 → ∞. 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.

Cited by

• Smooth concordance of cables of the figure-eight knot

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