Grope metrics on the knot concordance set

Powell, M. , Cochran, T. D. and Harvey, S. (2017) Grope metrics on the knot concordance set. Journal of Topology, 10(3), pp. 669-699. (doi: 10.1112/topo.12018)

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To a special type of grope embedded in 4-space, that we call a branch-symmetric grope, we associate a length function for each real number q ≥ 1. This gives rise to a family of pseudometrics dq, refining the slice genus metric, on the set of concordance classes of knots, as the infimum of the length function taken over all possible grope concordances between two knots. We investigate the properties of these metrics. The main theorem is that the topology induced by this metric on the knot concordance set is not discrete for all q > 1. The analogous statement for links also holds for q = 1. In addition, we translate much previous work on knot concordance into distance statements. In particular, we show that winding number zero satellite operators are contractions in many cases, and we give lower bounds on our metrics arising from knot signatures and higher order signatures. This gives further evidence in favor of the conjecture that the knot concordance group has a fractal structure.

Item Type:Articles
Glasgow Author(s) Enlighten ID:Powell, Dr Mark
Authors: Powell, M., Cochran, T. D., and Harvey, S.
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Journal Name:Journal of Topology
ISSN (Online):1753-8424
Published Online:07 June 2017

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