Rice University

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Seminar

Mathematics

Speaker: Lisa Piccirillo
University of Texas at Austin

Topology Seminar: Knot traces and concordance

Monday, March 27, 2017
4:00 PM  to 5:00 PM

227  Herman Brown Hall
Rice University
6100 Main St
Houston, Texas, USA

A classical conjecture of Akbulut and Kirby asserted that if a pair of knots $K_0$ and $K_1$ had homeomorphic zero surgeries then the knots should be (smoothly) concordant. Yasui disproved this in 2015 but the counterexamples he provided prompted Abe assert a corrected conjecture; If the 4-manifolds obtained by attaching a 2-handle to $B^4$ along $K_i$ with framing zero are diffeomorphic then the knots should be concordant. We disprove this by first giving a new method for constructing pairs of $K_i$ so that the associated four manifolds are diffeomorphic, then using the d-invariants of Heegaard Floer homology to obstruct some such $K_0$ and $K_1$ from being concordant. As a consequence we show that there exist invertible maps on the smooth concordance group which are induced via the satellite construction and with the property that $P(K)$ is not always concordant to $P(U)\# K$. This work is joint with Allison Miller.

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