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Transactions of the American Mathematical Society

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Cosmetic surgery in L-spaces and nugatory crossings

Authors: Tye Lidman and Allison H. Moore
Journal: Trans. Amer. Math. Soc. 369 (2017), 3639-3654
MSC (2010): Primary 57M25, 57M27
Published electronically: October 13, 2016
MathSciNet review: 3605982
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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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Additional Information

Tye Lidman
Affiliation: Institute for Advanced Study, Einstein Drive, Princeton, New Jersey 08540
Address at time of publication: Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695

Allison H. Moore
Affiliation: Department of Mathematics, Rice University, 6100 Main Street, Houston, Texas 77005
Address at time of publication: Department of Mathematics, University of California at Davis, One Shields Avenue, Davis, California 95616

Received by editor(s): July 19, 2015
Received by editor(s) in revised form: September 26, 2015
Published electronically: October 13, 2016
Additional Notes: The first author was partially supported by NSF RTG grant DMS-1148490.
The second author was partially supported by NSF grant DMS-1148609.
Article copyright: © Copyright 2016 American Mathematical Society

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