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Cosmetic crossing changesGeometry/Topology
|Speaker:||Allison Moore, UC Davis|
|Start time:||Wed, Sep 28 2016, 3:10PM|
A classic problem in knot theory is the cosmetic crossing conjecture, which asserts that the only crossing changes which preserve the isotopy class of a knot are nugatory crossing changes. Previously, the knots known to satisfy this conjecture included two-bridge and fibered knots. I will show that knots with branched double covers that are L-spaces also satisfy the cosmetic crossing conjecture, provided that the first singular homology of the branched double cover decomposes into summands of square-free order. The proof relies on the surgery characterization of the unknot, a tool coming from Floer homology, along with the G-equivariant Dehn's Lemma. I also plan to mention some potential generalizations to band surgery and subsequent applications. This is joint work with Lidman.