Cosmetic surgery in L-spaces and nugatory crossings

Tye Lidman; Allison H. Moore

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arXiv · 1507.00699

Cosmetic surgery in L-spaces and nugatory crossings

Tye Lidman·Allison H. Moore

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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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BibTeX RIS

Tye Lidman, Allison H. Moore. 2015-07-02. Cosmetic surgery in L-spaces and nugatory crossings. https://arxiv.org/abs/1507.00699

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