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3-braid knots do not admit purely cosmetic surgeries

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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… Click to show full abstract

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 Hanselman [5] 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.

Keywords: admit purely; purely cosmetic; knots admit; cosmetic surgeries; braid knots

Journal Title: Acta Mathematica Hungarica
Year Published: 2020

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