Dear Topologists and Friends of Topology,
We will have a seminar this Thursday (MSCS 514 at 3:30) given by Shelly Harvey, from Rice University.
Title: Metric Aspects of Knot Concordance
Abstract: We are interested in the set of knot up to concordance, denoted C. C is an abelian group but its structure is not very well understood. We propose a new approach to understanding C, namely considering C as a metric space on which there exists many natural operators. Since one example of such an operator is connected-sum with a fixed knot, this approach is arguably more general than focusing on C as an abelian group. In fact, it was previously suggested by the authors along with C. Leidy that C is a fractal space and the proposed self-similarities of C are classical satellite operations. Very recently, Cochran-Davis-Ray proved that many of these satellite operators (strong winding number 1) are indeed injective, modulo the smooth 4-dimensional Poincare Conjecture . We show that that these operators are, in fact, isometric embeddings while winding number zero satellite operators are, by contrast, approximate contractions. This is joint work with Tim Cochran.