Geometry & Topology

Knot commensurability and the Berge conjecture

Michel Boileau, Steven Boyer, Radu Cebanu, and Genevieve S Walsh

Full-text: Open access

Abstract

We investigate commensurability classes of hyperbolic knot complements in the generic case of knots without hidden symmetries. We show that such knot complements which are commensurable are cyclically commensurable, and that there are at most 3 hyperbolic knot complements in a cyclic commensurability class. Moreover if two hyperbolic knots have cyclically commensurable complements, then they are fibred with the same genus and are chiral. A characterization of cyclic commensurability classes of complements of periodic knots is also given. In the nonperiodic case, we reduce the characterization of cyclic commensurability classes to a generalization of the Berge conjecture.

Article information

Source
Geom. Topol., Volume 16, Number 2 (2012), 625-664.

Dates
Received: 8 February 2011
Accepted: 27 November 2011
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513732405

Digital Object Identifier
doi:10.2140/gt.2012.16.625

Mathematical Reviews number (MathSciNet)
MR2928979

Zentralblatt MATH identifier
1258.57001

Subjects
Primary: 57M10: Covering spaces 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}

Keywords
hyperbolic knot commensurability

Citation

Boileau, Michel; Boyer, Steven; Cebanu, Radu; Walsh, Genevieve S. Knot commensurability and the Berge conjecture. Geom. Topol. 16 (2012), no. 2, 625--664. doi:10.2140/gt.2012.16.625. https://projecteuclid.org/euclid.gt/1513732405


Export citation

References

  • I R Aitchison, J H Rubinstein, Combinatorial cubings, cusps, and the dodecahedral knots, from: “Topology '90 (Columbus, OH, 1990)”, (B Apanasov, W D Neumann, A W Reid, L Siebenmann, editors), Ohio State Univ. Math. Res. Inst. Publ. 1, de Gruyter, Berlin (1992) 17–26
  • I R Aitchison, J H Rubinstein, Geodesic surfaces in knot complements, Experiment. Math. 6 (1997) 137–150
  • M A Armstrong, The fundamental group of the orbit space of a discontinuous group, Proc. Cambridge Philos. Soc. 64 (1968) 299–301
  • J Berge, The knots in $D\sp 2\times S\sp 1$ which have nontrivial Dehn surgeries that yield $D\sp 2\times S\sp 1$, Topology Appl. 38 (1991) 1–19
  • M Boileau, B Leeb, J Porti, Geometrization of $3$–dimensional orbifolds, Ann. of Math. 162 (2005) 195–290
  • F Bonahon, J-P Otal, Scindements de Heegaard des espaces lenticulaires, Ann. Sci. École Norm. Sup. 16 (1983) 451–466
  • D Boyd, The A–polynomials of families of symmetric knots, Lecture notes, PIMS-MSRI conference on Knots and Manifolds (University of British Columbia, Vancouver) (2004) Available at \setbox0\makeatletter\@url http://www.math.ubc.ca/~boyd/Apoly.symm.pdf {\unhbox0
  • K S Brown, Trees, valuations, and the Bieri–Neumann–Strebel invariant, Invent. Math. 90 (1987) 479–504
  • D Calegari, N M Dunfield, Commensurability of $1$–cusped hyperbolic $3$–manifolds, Trans. Amer. Math. Soc. 354 (2002) 2955–2969
  • M Culler, C M Gordon, J Luecke, P B Shalen, Dehn surgery on knots, Ann. of Math. 125 (1987) 237–300
  • D Gabai, Foliations and the topology of $3$–manifolds, J. Differential Geom. 18 (1983) 445–503
  • D Gabai, Foliations and the topology of $3$–manifolds. II, J. Differential Geom. 26 (1987) 461–478
  • D Gabai, Surgery on knots in solid tori, Topology 28 (1989) 1–6
  • D Gabai, $1$–bridge braids in solid tori, Topology Appl. 37 (1990) 221–235
  • R E Gompf, ${\rm Spin}\sp c$–structures and homotopy equivalences, Geom. Topol. 1 (1997) 41–50
  • F González-Acuña, W C Whitten, Imbeddings of three-manifold groups, Mem. Amer. Math. Soc. 99, no. 474, Amer. Math. Soc. (1992)
  • O Goodman, D Heard, C Hodgson, Commensurators of cusped hyperbolic manifolds, Experiment. Math. 17 (2008) 283–306
  • C M Gordon, Dehn surgery and satellite knots, Trans. Amer. Math. Soc. 275 (1983) 687–708
  • C M Gordon, J Luecke, Knots are determined by their complements, J. Amer. Math. Soc. 2 (1989) 371–415
  • A Hatcher, Notes on basic $3$–manifold Topology, online book (2007) Available at \setbox0\makeatletter\@url http://www.math.cornell.edu/~hatcher/3M/3Mdownloads.html {\unhbox0
  • N Hoffman, Commensurability classes containing three knot complements, Algebr. Geom. Topol. 10 (2010) 663–677
  • A Juhász, Holomorphic discs and sutured manifolds, Algebr. Geom. Topol. 6 (2006) 1429–1457
  • A Juhász, Floer homology and surface decompositions, Geom. Topol. 12 (2008) 299–350
  • S Kojima, Isometry transformations of hyperbolic $3$–manifolds, Topology Appl. 29 (1988) 297–307
  • P B Kronheimer, T S Mrowka, Dehn surgery, the fundamental group and SU$(2)$, Math. Res. Lett. 11 (2004) 741–754
  • M L Macasieb, T W Mattman, Commensurability classes of $(-2,3,n)$ pretzel knot complements, Algebr. Geom. Topol. 8 (2008) 1833–1853
  • J Morgan, G Tian, Ricci flow and the Poincaré conjecture, Clay Math. Monogr. 3, Amer. Math. Soc. (2007)
  • W D Neumann, Kleinian groups generated by rotations, from: “Groups–-Korea '94 (Pusan)”, (A C Kim, D L Johnson, editors), de Gruyter, Berlin (1995) 251–256
  • W D Neumann, A W Reid, Arithmetic of hyperbolic manifolds, from: “Topology '90 (Columbus, OH, 1990)”, (B Apanasov, W D Neumann, A W Reid, L Siebenmann, editors), Ohio State Univ. Math. Res. Inst. Publ. 1, de Gruyter, Berlin (1992) 273–310
  • Y Ni, Knot Floer homology detects fibred knots, Invent. Math. 170 (2007) 577–608
  • Y Ni, Link Floer homology detects the Thurston norm, Geom. Topol. 13 (2009) 2991–3019
  • P Ozsváth, Z Szabó, Knot Floer homology and rational surgeries
  • P Ozsváth, Z Szabó, Holomorphic disks and knot invariants, Adv. Math. 186 (2004) 58–116
  • P Ozsváth, Z Szabó, Holomorphic disks and three-manifold invariants: properties and applications, Ann. of Math. 159 (2004) 1159–1245
  • P Ozsváth, Z Szabó, Holomorphic disks and topological invariants for closed three-manifolds, Ann. of Math. 159 (2004) 1027–1158
  • P Ozsváth, Z Szabó, On knot Floer homology and lens space surgeries, Topology 44 (2005) 1281–1300
  • P Ozsváth, Z Szabó, Holomorphic disks, link invariants and the multi-variable Alexander polynomial, Algebr. Geom. Topol. 8 (2008) 615–692
  • J Rasmussen, Lens space surgeries and L-space homology spheres
  • A W Reid, Arithmeticity of knot complements, J. London Math. Soc. 43 (1991) 171–184
  • A W Reid, G S Walsh, Commensurability classes of 2-bridge knot complements, Algebr. Geom. Topol. 8 (2008) 1031–1057
  • T Sakai, Geodesic knots in a hyperbolic $3$–manifold, Kobe J. Math. 8 (1991) 81–87
  • R E Schwartz, The quasi-isometry classification of rank one lattices, Inst. Hautes Études Sci. Publ. Math. (1995) 133–168
  • J Stallings, On fibering certain $3$–manifolds, from: “Topology of $3$–manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961)”, (M K Fort, editor), Prentice-Hall, Englewood Cliffs, N.J. (1962) 95–100
  • J Stallings, Constructions of fibred knots and links, from: “Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, CA, 1976), Part 2”, (R J Milgram, editor), Proc. Sympos. Pure Math. XXXII, Amer. Math. Soc. (1978) 55–60
  • W P Thurston, A norm for the homology of $3$–manifolds, Mem. Amer. Math. Soc. 59, no. 339, Amer. Math. Soc. (1986)
  • V Turaev, Torsion invariants of ${\rm Spin}\sp c$–structures on $3$–manifolds, Math. Res. Lett. 4 (1997) 679–695
  • C A Weibel, An introduction to homological algebra, Cambridge Studies in Adv. Math. 38, Cambridge Univ. Press (1994)