## Algebraic & Geometric Topology

### The geometry of the knot concordance space

#### Abstract

Most of the $5 0$ years of study of the set of knot concordance classes, $ℂ$, has focused on its structure as an abelian group. Here we take a different approach, namely we study $ℂ$ 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 $ℂ$ admits no quasi-isometric embedding into a finite product of (Gromov) hyperbolic spaces. We show that every satellite operator is a quasihomomorphism $P : ℂ → ℂ$. 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 $ℂ$ is a fractal space. We establish various other results about the large-scale geometry of arbitrary satellite operators.

#### Article information

Source
Algebr. Geom. Topol., Volume 18, Number 5 (2018), 2509-2540.

Dates
Revised: 15 May 2017
Accepted: 15 July 2017
First available in Project Euclid: 30 August 2018

https://projecteuclid.org/euclid.agt/1535594416

Digital Object Identifier
doi:10.2140/agt.2018.18.2509

Mathematical Reviews number (MathSciNet)
MR3848393

Zentralblatt MATH identifier
06935814

#### Citation

Cochran, Tim D; Harvey, Shelly. The geometry of the knot concordance space. Algebr. Geom. Topol. 18 (2018), no. 5, 2509--2540. doi:10.2140/agt.2018.18.2509. https://projecteuclid.org/euclid.agt/1535594416

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