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Grope metrics on the knot concordance set

Cochran, Tim D.; Harvey, Shelly; Powell, Mark

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Authors

Tim D. Cochran

Shelly Harvey

Mark Powell



Abstract

To a special type of grope embedded in 4-space, that we call a branchsymmetric grope, we associate a length function for each real number q ≥ 1. This gives rise to a family of pseudo-metrics d q , 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.

Citation

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

Journal Article Type Article
Acceptance Date Feb 13, 2017
Online Publication Date Jun 7, 2017
Publication Date Sep 1, 2017
Deposit Date Oct 3, 2017
Publicly Available Date Oct 3, 2017
Journal Journal of Topology
Print ISSN 1753-8416
Electronic ISSN 1753-8424
Publisher Wiley
Peer Reviewed Peer Reviewed
Volume 10
Issue 3
Pages 669-699
DOI https://doi.org/10.1112/topo.12018
Related Public URLs https://arxiv.org/abs/1512.06897

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Copyright Statement
This is a pre-copyedited, author-produced PDF of an article accepted for publication in the Journal of Topology following peer review. The version of record is available online at: https://doi.org/10.1112/topo.12018




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