Topology Seminar (Main Talk): Cosmetic crossing changes

Seminar | April 26 | 4:10-5 p.m. | 3 Evans Hall

 Allison Moore, UC Davis

 Department of Mathematics

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 related problems involving the operation of band surgery and subsequent applications to problems in DNA topology. This is joint work with Lidman.