We determine the lens spaces that arise by integer Dehn surgery along a knot in the three-sphere. Specifically, if surgery along a knot produces a lens space, then there exists an equivalent surgery along a Berge knot with the same knot Floer homology groups. This leads to sharp information about the genus of such a knot. The arguments rely on tools from Floer homology and lattice theory, and are primarily combinatorial in nature.
Abstract. We exhibit the first examples of links which are homologically thin but not quasi-alternating. To show that they are not quasi-alternating, we argue that none of their branched double-covers bounds a negative definite 4-manifold with non-torsion H1. Using this method, we also complete the determination of the quasi-alternating pretzel links.
We determine the (smooth) concordance order of the 3-stranded pretzel knots P (p, q, r) with p, q, r odd. We show that each one of finite order is, in fact, ribbon, thereby proving the slice-ribbon conjecture for this family of knots. As corollaries we give new proofs of results first obtained
We establish a characterization of alternating links in terms of definite
spanning surfaces. We apply it to obtain a new proof of Tait's conjecture that
reduced alternating diagrams of the same link have the same crossing number and
writhe. We also deduce a result of Banks and Hirasawa-Sakuma about Seifert
surfaces for special alternating links. The appendix, written by Juh\'asz and
Lackenby, applies the characterization to derive an exponential time algorithm
for alternating knot recognition.Comment: 11 pages, 1 figur
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.Partially supported by an NSF Post-doctoral Fellowship.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.