×

Knot commensurability and the Berge conjecture. (English) Zbl 1258.57001

This paper is about knot commensurability. Two knots in the 3-sphere are called commensurable if their complements are commensurable, i.e. have homeomorphic finite sheeted covers. A. W. Reid [J. Lond. Math. Soc., II. Ser. 43, No. 1, 171–184 (1991; Zbl 0847.57013)] has shown that the figure eight knot is the only arithmetic knot in \(S^3\), which implies that it is the unique knot in its commensurability class.
Reid and Walsh have conjectured that when a knot \(K\) is hyperbolic, there are at most three distinct knots in the commensurability class of \(K\). In the paper under review, this conjecture is proved for so-called flexible knots, a condition which is expected to be generic. Another result states that (1) knots without hidden symmetries which are commensurable are cyclically commensurable, and (2) a cyclic commensurability class contains at most three hyperbolic knot complements.
There are other results concerning fibred knots and periodic knots.

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M10 Covering spaces and low-dimensional topology

Citations:

Zbl 0847.57013
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] I R Aitchison, J H Rubinstein, Combinatorial cubings, cusps, and the dodecahedral knots (editors B Apanasov, W D Neumann, A W Reid, L Siebenmann), Ohio State Univ. Math. Res. Inst. Publ. 1, de Gruyter (1992) 17 · Zbl 0773.57010
[2] I R Aitchison, J H Rubinstein, Geodesic surfaces in knot complements, Experiment. Math. 6 (1997) 137 · Zbl 0891.57017
[3] M A Armstrong, The fundamental group of the orbit space of a discontinuous group, Proc. Cambridge Philos. Soc. 64 (1968) 299 · Zbl 0159.53002
[4] J Berge, The knots in \(D^2\times S^1\) which have nontrivial Dehn surgeries that yield \(D^2\times S^1\), Topology Appl. 38 (1991) 1 · Zbl 0725.57001
[5] M Boileau, B Leeb, J Porti, Geometrization of \(3\)-dimensional orbifolds, Ann. of Math. 162 (2005) 195 · Zbl 1087.57009
[6] F Bonahon, J P Otal, Scindements de Heegaard des espaces lenticulaires, Ann. Sci. École Norm. Sup. 16 (1983) 451 · Zbl 0545.57002
[7] 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)
[8] K S Brown, Trees, valuations, and the Bieri-Neumann-Strebel invariant, Invent. Math. 90 (1987) 479 · Zbl 0663.20033
[9] D Calegari, N M Dunfield, Commensurability of \(1\)-cusped hyperbolic \(3\)-manifolds, Trans. Amer. Math. Soc. 354 (2002) 2955 · Zbl 0986.57005
[10] M Culler, C M Gordon, J Luecke, P B Shalen, Dehn surgery on knots, Ann. of Math. 125 (1987) 237 · Zbl 0633.57006
[11] D Gabai, Foliations and the topology of \(3\)-manifolds, J. Differential Geom. 18 (1983) 445 · Zbl 0539.57013
[12] D Gabai, Foliations and the topology of \(3\)-manifolds. II, J. Differential Geom. 26 (1987) 461 · Zbl 0627.57012
[13] D Gabai, Surgery on knots in solid tori, Topology 28 (1989) 1 · Zbl 0678.57004
[14] D Gabai, \(1\)-bridge braids in solid tori, Topology Appl. 37 (1990) 221 · Zbl 0817.57006
[15] R E Gompf, \(\mathrm{Spin}^c\)-structures and homotopy equivalences, Geom. Topol. 1 (1997) 41 · Zbl 0886.57021
[16] F González-Acuña, W C Whitten, Imbeddings of three-manifold groups, Mem. Amer. Math. Soc. 99, no. 474, Amer. Math. Soc. (1992) · Zbl 0756.57002
[17] O Goodman, D Heard, C Hodgson, Commensurators of cusped hyperbolic manifolds, Experiment. Math. 17 (2008) 283 · Zbl 1338.57016
[18] C M Gordon, Dehn surgery and satellite knots, Trans. Amer. Math. Soc. 275 (1983) 687 · Zbl 0519.57005
[19] C M Gordon, J Luecke, Knots are determined by their complements, J. Amer. Math. Soc. 2 (1989) 371 · Zbl 0672.57009
[20] A Hatcher, Notes on basic \(3\)-manifold Topology, online book (2007)
[21] N Hoffman, Commensurability classes containing three knot complements, Algebr. Geom. Topol. 10 (2010) 663 · Zbl 1188.57001
[22] A Juhász, Holomorphic discs and sutured manifolds, Algebr. Geom. Topol. 6 (2006) 1429 · Zbl 1129.57039
[23] A Juhász, Floer homology and surface decompositions, Geom. Topol. 12 (2008) 299 · Zbl 1167.57005
[24] S Kojima, Isometry transformations of hyperbolic \(3\)-manifolds, Topology Appl. 29 (1988) 297 · Zbl 0654.57006
[25] P B Kronheimer, T S Mrowka, Dehn surgery, the fundamental group and SU\((2)\), Math. Res. Lett. 11 (2004) 741 · Zbl 1084.57006
[26] M L Macasieb, T W Mattman, Commensurability classes of \((-2,3,n)\) pretzel knot complements, Algebr. Geom. Topol. 8 (2008) 1833 · Zbl 1162.57005
[27] J Morgan, G Tian, Ricci flow and the Poincaré conjecture, Clay Math. Monogr. 3, Amer. Math. Soc. (2007) · Zbl 1179.57045
[28] W D Neumann, Kleinian groups generated by rotations (editors A C Kim, D L Johnson), de Gruyter (1995) 251 · Zbl 0872.20045
[29] W D Neumann, A W Reid, Arithmetic of hyperbolic manifolds (editors B Apanasov, W D Neumann, A W Reid, L Siebenmann), Ohio State Univ. Math. Res. Inst. Publ. 1, de Gruyter (1992) 273 · Zbl 0777.57007
[30] Y Ni, Knot Floer homology detects fibred knots, Invent. Math. 170 (2007) 577 · Zbl 1138.57031
[31] Y Ni, Link Floer homology detects the Thurston norm, Geom. Topol. 13 (2009) 2991 · Zbl 1203.57005
[32] P Ozsváth, Z Szabó, Knot Floer homology and rational surgeries · Zbl 1226.57044
[33] P Ozsváth, Z Szabó, Holomorphic disks and knot invariants, Adv. Math. 186 (2004) 58 · Zbl 1062.57019
[34] P Ozsváth, Z Szabó, Holomorphic disks and three-manifold invariants: properties and applications, Ann. of Math. 159 (2004) 1159 · Zbl 1081.57013
[35] P Ozsváth, Z Szabó, Holomorphic disks and topological invariants for closed three-manifolds, Ann. of Math. 159 (2004) 1027 · Zbl 1073.57009
[36] P Ozsváth, Z Szabó, On knot Floer homology and lens space surgeries, Topology 44 (2005) 1281 · Zbl 1077.57012
[37] P Ozsváth, Z Szabó, Holomorphic disks, link invariants and the multi-variable Alexander polynomial, Algebr. Geom. Topol. 8 (2008) 615 · Zbl 1144.57011
[38] J Rasmussen, Lens space surgeries and L-space homology spheres
[39] A W Reid, Arithmeticity of knot complements, J. London Math. Soc. 43 (1991) 171 · Zbl 0847.57013
[40] A W Reid, G S Walsh, Commensurability classes of 2-bridge knot complements, Algebr. Geom. Topol. 8 (2008) 1031 · Zbl 1154.57001
[41] T Sakai, Geodesic knots in a hyperbolic \(3\)-manifold, Kobe J. Math. 8 (1991) 81 · Zbl 0749.57003
[42] R E Schwartz, The quasi-isometry classification of rank one lattices, Inst. Hautes Études Sci. Publ. Math. (1995) 133 · Zbl 0852.22010
[43] J Stallings, On fibering certain \(3\)-manifolds (editor M K Fort), Prentice-Hall (1962) 95 · Zbl 1246.57049
[44] J Stallings, Constructions of fibred knots and links (editor R J Milgram), Proc. Sympos. Pure Math. XXXII, Amer. Math. Soc. (1978) 55 · Zbl 0394.57007
[45] W P Thurston, A norm for the homology of \(3\)-manifolds, Mem. Amer. Math. Soc. 59, no. 339, Amer. Math. Soc. (1986) · Zbl 0585.57006
[46] V Turaev, Torsion invariants of \(\mathrm{Spin}^c\)-structures on \(3\)-manifolds, Math. Res. Lett. 4 (1997) 679 · Zbl 0891.57019
[47] C A Weibel, An introduction to homological algebra, Cambridge Studies in Adv. Math. 38, Cambridge Univ. Press (1994) · Zbl 0797.18001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.