2:00 pm Monday, October 27, 2014
Topology Seminar: A rational valued metric on the knot concordance group coming from gropes by Shelly Harvey (Rice University) in RLM 12.166
Most of the 50-year history of the study of the set of smooth knot concordance classes, C, has focused on its structure as an abelian group. Recently Tim Cochran and I took a different approach, namely we studied C as a metric space (with the slice genus metric or the homology metrics) admitting many natural geometric operators, especially satellite operators. The hope was to give evidence that the knot concordance is a fractal space. However, both of these metrics are integer valued metrics and so induce the discrete topology. Here (with Mark Powell) we define a family of metrics, called the q-grope metrics (0 q = 1), which take values in Q. We will show that there are sequences of knots whose q-norms get arbitrarily small for q1. We will also show that for any winding number 0 satellite operator , R:C - C, there is an interval for q such that R is a contraction. If there is time, we will describe another Q-valued metric based on kinky handles. This is joint work with Tim Cochran and Mark Powell. Submitted by
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