Tim D. Cochran
Grope metrics on the knot concordance set
Cochran, Tim D.; Harvey, Shelly; Powell, Mark
Authors
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 |
Files
Accepted Journal Article
(267 Kb)
PDF
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
You might also like
Embedded surfaces with infinite cyclic knot group
(2023)
Journal Article
The Z-Genus of Boundary Links
(2022)
Journal Article
Four-manifolds up to connected sum with complex projective planes
(2022)
Journal Article
Embedding spheres in knot traces
(2021)
Journal Article
Characterisation of homotopy ribbon discs
(2021)
Journal Article
Downloadable Citations
About Durham Research Online (DRO)
Administrator e-mail: dro.admin@durham.ac.uk
This application uses the following open-source libraries:
SheetJS Community Edition
Apache License Version 2.0 (http://www.apache.org/licenses/)
PDF.js
Apache License Version 2.0 (http://www.apache.org/licenses/)
Font Awesome
SIL OFL 1.1 (http://scripts.sil.org/OFL)
MIT License (http://opensource.org/licenses/mit-license.html)
CC BY 3.0 ( http://creativecommons.org/licenses/by/3.0/)
Powered by Worktribe © 2024
Advanced Search