Commensurability classes of 2–bridge knot complements

Reid, Alan W¹ ; Walsh, Genevieve S²

¹Department of Mathematics, University of Texas, Austin, TX 78712, USA
²Department of Mathematics, Tufts University, Medford, MA 02155, USA

Algebraic and Geometric Topology, Tome 8 (2008) no. 2, pp. 1031-1057

Résumé

We show that a hyperbolic 2–bridge knot complement is the unique knot complement in its commensurability class. We also discuss constructions of commensurable hyperbolic knot complements and put forth a conjecture on the number of hyperbolic knot complements in a commensurability class.

Détail

DOI : 10.2140/agt.2008.8.1031

Keywords: commensurability, hyperbolic knot complement, 2-bridge knot

Bibliographie

[1] I R Aitchison, J H Rubinstein, Combinatorial cubings, cusps, and the dodecahedral knots, from: "Topology '90 (Columbus, OH, 1990)", Ohio State Univ. Math. Res. Inst. Publ. 1, de Gruyter (1992) 17

[2] J Berge, Some knots with surgeries yielding lens spaces, to appear in the proceedings of the Cassonfest

[3] M Boileau, S Boyer, G S Walsh, On commensurability of knot complements, preprint available at http://www.tufts.edu/ gwalsh01/

[4] S Boyer, X Zhang, Finite Dehn surgery on knots, J. Amer. Math. Soc. 9 (1996) 1005

[5] M Culler, C M Gordon, J Luecke, P B Shalen, Dehn surgery on knots, Ann. of Math. $(2)$ 125 (1987) 237

[6] W D Dunbar, Hierarchies for $3$–orbifolds, Topology Appl. 29 (1988) 267

[7] F González-Acuña, W C Whitten, Imbeddings of three-manifold groups, Mem. Amer. Math. Soc. 99 (1992)

[8] O Goodman, D Heard, C Hodgson, Commensurators of cusped hyperbolic manifolds

[9] C M Gordon, Dehn filling: a survey, from: "Knot theory (Warsaw, 1995)", Banach Center Publ. 42, Polish Acad. Sci. (1998) 129

[10] J Hoste, P D Shanahan, Trace fields of twist knots, J. Knot Theory Ramifications 10 (2001) 625

[11] H Koch, Number theory, Graduate Studies in Math. 24, Amer. Math. Soc. (2000)

[12] M L Macasieb, T W Mattman, Commensurability classes of $(-2,3,n)$ pretzel knot complements

[13] C Maclachlan, A W Reid, The arithmetic of hyperbolic $3$–manifolds, Graduate Texts in Math. 219, Springer (2003)

[14] G A Margulis, Discrete subgroups and ergodic theory, from: "Number theory, trace formulas and discrete groups (Oslo, 1987)", Academic Press (1989) 377

[15] J Milnor, Groups which act on $S^n$ without fixed points, Amer. J. Math. 79 (1957) 623

[16] W D Neumann, Commensurability and virtual fibration for graph manifolds, Topology 36 (1997) 355

[17] W D Neumann, A W Reid, Arithmetic of hyperbolic manifolds, from: "Topology '90 (Columbus, OH, 1990)", Ohio State Univ. Math. Res. Inst. Publ. 1, de Gruyter (1992) 273

[18] P Ozsváth, Z Szabó, On knot Floer homology and lens space surgeries, Topology 44 (2005) 1281

[19] A W Reid, A note on trace-fields of Kleinian groups, Bull. London Math. Soc. 22 (1990) 349

[20] A W Reid, Arithmeticity of knot complements, J. London Math. Soc. $(2)$ 43 (1991) 171

[21] R Riley, Parabolic representations of knot groups. I, Proc. London Math. Soc. $(3)$ 24 (1972) 217

[22] R Riley, Seven excellent knots, from: "Low-dimensional topology (Bangor, 1979)", London Math. Soc. Lecture Note Ser. 48, Cambridge Univ. Press (1982) 81

[23] R Riley, Parabolic representations and symmetries of the knot $9_{32}$, from: "Computers in geometry and topology (Chicago, IL, 1986)", Lecture Notes in Pure and Appl. Math. 114, Dekker (1989) 297

[24] M Sakuma, The geometries of spherical Montesinos links, Kobe J. Math. 7 (1990) 167

[25] R E Schwartz, The quasi-isometry classification of rank one lattices, Inst. Hautes Études Sci. Publ. Math. 82 (1995) 133

[26] M Suzuki, Group theory. I, Grundlehren series 247, Springer (1982)

[27] M O Takahashi, Two-bridge knots have property $\mathrm{P}$, Mem. Amer. Math. Soc. 29 (1981)

[28] S C Wang, Y Q Wu, Any knot complement covers at most one knot complement, Pacific J. Math. 158 (1993) 387

[29] S C Wang, Q Zhou, Symmetry of knots and cyclic surgery, Trans. Amer. Math. Soc. 330 (1992) 665