Cosmetic surgery in L-spaces and nugatory crossings

Lidman, Tye; Moore, Allison

doi:10.1090/tran/6839

Cosmetic surgery in L-spaces and nugatory crossings
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by Tye Lidman and Allison H. Moore PDF

Trans. Amer. Math. Soc. 369 (2017), 3639-3654 Request permission

Abstract:

The cosmetic crossing conjecture (also known as the “nugatory crossing conjecture”) asserts that the only crossing changes that preserve the oriented isotopy class of a knot in the 3-sphere are nugatory. We use the Dehn surgery characterization of the unknot to prove this conjecture for knots in integer homology spheres whose branched double covers are L-spaces satisfying a homological condition. This includes as a special case all alternating and quasi-alternating knots with square-free determinant. As an application, we prove the cosmetic crossing conjecture holds for all knots with at most nine crossings and provide new examples of knots, including pretzel knots, non-arborescent knots and symmetric unions for which the conjecture holds.

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