Algebraic & Geometric Topology

The geometry of the knot concordance space

Tim D Cochran and Shelly Harvey

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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.

Article information

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

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

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx] 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 57M27: Invariants of knots and 3-manifolds

knot concordance satellite operator metric fractal


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.

Export citation


  • S Akbulut, R Kirby, Mazur manifolds, Michigan Math. J. 26 (1979) 259–284
  • S Akbulut, K Yasui, Corks, plugs and exotic structures, J. Gökova Geom. Topol. 2 (2008) 40–82
  • L Bartholdi, R Grigorchuk, V Nekrashevych, From fractal groups to fractal sets, from “Fractals in Graz 2001” (P Grabner, W Woess, editors), Birkhäuser, Basel (2003) 25–118
  • S Boyer, Shake-slice knots and smooth contractible $4$–manifolds, Math. Proc. Cambridge Philos. Soc. 98 (1985) 93–106
  • M Brandenbursky, J K\kedra, Concordance group and stable commutator length in braid groups, Algebr. Geom. Topol. 15 (2015) 2861–2886
  • A Casson, M Freedman, Atomic surgery problems, from “Four-manifold theory” (C Gordon, R Kirby, editors), Contemp. Math. 35, Amer. Math. Soc., Providence, RI (1984) 181–199
  • J C Cha, The structure of the rational concordance group of knots, Mem. Amer. Math. Soc. 885, Amer. Math. Soc., Providence, RI (2007)
  • T D Cochran, C W Davis, Counterexamples to Kauffman's conjectures on slice knots, Adv. Math. 274 (2015) 263–284
  • T D Cochran, C W Davis, A Ray, Injectivity of satellite operators in knot concordance, J. Topol. 7 (2014) 948–964
  • T D Cochran, B D Franklin, M Hedden, P D Horn, Knot concordance and homology cobordism, Proc. Amer. Math. Soc. 141 (2013) 2193–2208
  • T D Cochran, R E Gompf, Applications of Donaldson's theorems to classical knot concordance, homology $3$–spheres and property $P$, Topology 27 (1988) 495–512
  • T D Cochran, S Harvey, C Leidy, $2$–torsion in the $n$–solvable filtration of the knot concordance group, Proc. Lond. Math. Soc. 102 (2011) 257–290
  • T D Cochran, S Harvey, C Leidy, Primary decomposition and the fractal nature of knot concordance, Math. Ann. 351 (2011) 443–508
  • J Collins, On the concordance orders of knots, PhD thesis, University of Edinburgh (2012)
  • R H Fox, J W Milnor, Singularities of $2$–spheres in $4$–space and equivalence of knots, Bull. Amer. Math. Soc. 63 (1957) 406
  • R H Fox, J W Milnor, Singularities of $2$–spheres in $4$–space and cobordism of knots, Osaka J. Math. 3 (1966) 257–267
  • M H Freedman, F Quinn, Topology of $4$–manifolds, Princeton Mathematical Series 39, Princeton Univ. Press (1990)
  • S L Harvey, Homology cobordism invariants and the Cochran–Orr–Teichner filtration of the link concordance group, Geom. Topol. 12 (2008) 387–430
  • M Hedden, P Kirk, Instantons, concordance, and Whitehead doubling, J. Differential Geom. 91 (2012) 281–319
  • C Kearton, The Milnor signatures of compound knots, Proc. Amer. Math. Soc. 76 (1979) 157–160
  • R C Kirby, editor, Problems in low-dimensional topology, from “Geometric topology, II” (W H Kazez, editor), AMS/IP Stud. Adv. Math. 2, Amer. Math. Soc., Providence, RI (1997) 35–473
  • P Kirk, C Livingston, Twisted knot polynomials: inversion, mutation and concordance, Topology 38 (1999) 663–671
  • A S Levine, Nonsurjective satellite operators and piecewise-linear concordance, Forum Math. Sigma 4 (2016) art. id. e34
  • W B R Lickorish, An introduction to knot theory, Graduate Texts in Mathematics 175, Springer (1997)
  • C Livingston, The slicing number of a knot, Algebr. Geom. Topol. 2 (2002) 1051–1060
  • C Livingston, The concordance genus of knots, Algebr. Geom. Topol. 4 (2004) 1–22
  • C Livingston, The stable 4-genus of knots, Algebr. Geom. Topol. 10 (2010) 2191–2202
  • S Naik, Casson–Gordon invariants of genus one knots and concordance to reverses, J. Knot Theory Ramifications 5 (1996) 661–677
  • A Ray, Satellite operators with distinct iterates in smooth concordance, Proc. Amer. Math. Soc. 143 (2015) 5005–5020
  • Y Sato, $3$–dimensional homology handles and minimal second Betti numbers of $4$–manifolds, Osaka J. Math. 35 (1998) 509–527
  • L R Taylor, On the genera of knots, from “Topology of low-dimensional manifolds” (R A Fenn, editor), Lecture Notes in Math. 722, Springer (1979) 144–154
  • A G Tristram, Some cobordism invariants for links, Proc. Cambridge Philos. Soc. 66 (1969) 251–264