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2018 The geometry of the knot concordance space
Tim D Cochran, Shelly Harvey
Algebr. Geom. Topol. 18(5): 2509-2540 (2018). DOI: 10.2140/agt.2018.18.2509


Most of the 5 0 years of study of the set of knot concordance classes, , has focused on its structure as an abelian group. Here we take a different approach, namely we study as a metric space admitting many natural geometric operators. We focus especially on the coarse geometry of satellite operators. We consider several knot concordance spaces, corresponding to different categories of concordance, and two different metrics. We establish the existence of quasi- n –flats for every n , implying that admits no quasi-isometric embedding into a finite product of (Gromov) hyperbolic spaces. We show that every satellite operator is a quasihomomorphism P : . We show that winding number one satellite operators induce quasi-isometries with respect to the metric induced by slice genus. We prove that strong winding number one patterns induce isometric embeddings for certain metrics. By contrast, winding number zero satellite operators are bounded functions and hence quasicontractions. These results contribute to the suggestion that is a fractal space. We establish various other results about the large-scale geometry of arbitrary satellite operators.


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Tim D Cochran. Shelly Harvey. "The geometry of the knot concordance space." Algebr. Geom. Topol. 18 (5) 2509 - 2540, 2018.


Received: 16 September 2014; Revised: 15 May 2017; Accepted: 15 July 2017; Published: 2018
First available in Project Euclid: 30 August 2018

zbMATH: 06935814
MathSciNet: MR3848393
Digital Object Identifier: 10.2140/agt.2018.18.2509

Primary: 20F65, 57M25, 57M27

Rights: Copyright © 2018 Mathematical Sciences Publishers


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Vol.18 • No. 5 • 2018
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