L-space surgeries, genus bounds, and the cabling conjecture
By J. Greene. 2010
61 citations recorded by the source corpus.
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.
Subfields: Genetics, Geometry and Topology, Neurology
View Paper: https://www.semanticscholar.org/paper/4e34c96cf084b70c98c27e84b92cc89b83ab01c8
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- The lens space realization problem (2010, methodology): “ulate the correction terms eectively towards this end. The key idea here is to use the correction terms in tandem with Donaldson’s Theorem. The result is an enhanced lattice embedding condition (cf. [17, 18, 19]). In order to state it, we rst require a combinatorial denition. Denition 1.5. A vector ”