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3-braid knots do not admit purely cosmetic surgeries. (English) Zbl 1499.57007

Summary: A pair of surgeries on a knot is called purely cosmetic if the pair of resulting 3-manifolds are homeomorphic as oriented manifolds. An outstanding conjecture predicts that no nontrivial knots admit any purely cosmetic surgeries. Recent work of J. Hanselman [“Heegaard Floer homology and cosmetic surgeries in \(S^3\)”, Preprint, arXiv:1906.06773] uses Heegaard Floer homology to obtain new obstructions for the existence of such surgeries. In this work, we apply those obstructions to show that (nontrivial) knots which arise as the closure of a 3-stranded braid do not admit any purely cosmetic surgeries.

MSC:

57K10 Knot theory
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References:

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