Commensurability of 1-cusped hyperbolic 3-manifolds
HTML articles powered by AMS MathViewer
- by Danny Calegari and Nathan M. Dunfield PDF
- Trans. Amer. Math. Soc. 354 (2002), 2955-2969 Request permission
Abstract:
We give examples of non-fibered hyperbolic knot complements in homology spheres that are not commensurable to fibered knot complements in homology spheres. In fact, we give many examples of knot complements in homology spheres where every commensurable knot complement in a homology sphere has non-monic Alexander polynomial.References
- I. R. Aitchison and J. H. Rubinstein, Combinatorial cubings, cusps, and the dodecahedral knots, Topology ’90 (Columbus, OH, 1990) Ohio State Univ. Math. Res. Inst. Publ., vol. 1, de Gruyter, Berlin, 1992, pp. 17–26. MR 1184399
- Robert Bieri, Walter D. Neumann, and Ralph Strebel, A geometric invariant of discrete groups, Invent. Math. 90 (1987), no. 3, 451–477. MR 914846, DOI 10.1007/BF01389175
- A. Borel, Commensurability classes and volumes of hyperbolic $3$-manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 8 (1981), no. 1, 1–33. MR 616899
- Kenneth S. Brown, Trees, valuations, and the Bieri-Neumann-Strebel invariant, Invent. Math. 90 (1987), no. 3, 479–504. MR 914847, DOI 10.1007/BF01389176
- Michel Boileau and Shicheng Wang, Non-zero degree maps and surface bundles over $S^1$, J. Differential Geom. 43 (1996), no. 4, 789–806. MR 1412685
- S. Boyer and X. Zhang, On Culler-Shalen seminorms and Dehn filling, Ann. of Math. (2) 148 (1998), no. 3, 737–801. MR 1670053, DOI 10.2307/121031
- Gerhard Burde and Heiner Zieschang, Knots, De Gruyter Studies in Mathematics, vol. 5, Walter de Gruyter & Co., Berlin, 1985. MR 808776
- Marc Culler, C. McA. Gordon, J. Luecke, and Peter B. Shalen, Dehn surgery on knots, Ann. of Math. (2) 125 (1987), no. 2, 237–300. MR 881270, DOI 10.2307/1971311
- Patrick J. Callahan, Martin V. Hildebrand, and Jeffrey R. Weeks, A census of cusped hyperbolic $3$-manifolds, Math. Comp. 68 (1999), no. 225, 321–332. With microfiche supplement. MR 1620219, DOI 10.1090/S0025-5718-99-01036-4
- Marc Culler and Peter B. Shalen, Varieties of group representations and splittings of $3$-manifolds, Ann. of Math. (2) 117 (1983), no. 1, 109–146. MR 683804, DOI 10.2307/2006973
- Nathan M. Dunfield, Alexander and Thurston norms of 3-manifolds fibering over the circle, Pacific J. Math 200 (2001), no. 1, 43–58, arXiv:math.GT/9908050.
- O. Goodman, Snap, http://www.ms.unimelb.edu.au/~snap/.
- Hugh M. Hilden, María Teresa Lozano, and José María Montesinos-Amilibia, On the arithmetic $2$-bridge knots and link orbifolds and a new knot invariant, J. Knot Theory Ramifications 4 (1995), no. 1, 81–114. MR 1321291, DOI 10.1142/S0218216595000053
- Craig D. Hodgson, G. Robert Meyerhoff, and Jeffrey R. Weeks, Surgeries on the Whitehead link yield geometrically similar manifolds, Topology ’90 (Columbus, OH, 1990) Ohio State Univ. Math. Res. Inst. Publ., vol. 1, de Gruyter, Berlin, 1992, pp. 195–206. MR 1184411
- A. Hatcher and W. Thurston, Incompressible surfaces in $2$-bridge knot complements, Invent. Math. 79 (1985), no. 2, 225–246. MR 778125, DOI 10.1007/BF01388971
- E. Kaltofen, Polynomial factorization, Computer Algebra (B. Buchberger, G. Collins, and R. Loos, eds.), Springer Verlag, Heidelberg, 2 ed., 1982, pp. 95–113.
- Erich Kaltofen, Polynomial factorization 1982–1986, Computers in mathematics (Stanford, CA, 1986) Lecture Notes in Pure and Appl. Math., vol. 125, Dekker, New York, 1990, pp. 285–309. MR 1068540
- William H. Kazez (ed.), Geometric topology, AMS/IP Studies in Advanced Mathematics, vol. 2, American Mathematical Society, Providence, RI; International Press, Cambridge, MA, 1997. MR 1470749, DOI 10.1090/amsip/002.2
- T. Venkatarayudu, The $7$-$15$ problem, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 531. MR 0000001, DOI 10.1090/gsm/058
- D. D. Long and A. W. Reid, Commensurability and the character variety, Math. Res. Lett. 6 (1999), no. 5-6, 581–591. MR 1739217, DOI 10.4310/MRL.1999.v6.n5.a11
- Alexander Lubotzky, Subgroup growth and congruence subgroups, Invent. Math. 119 (1995), no. 2, 267–295. MR 1312501, DOI 10.1007/BF01245183
- Curtis T. McMullen, The Alexander polynomial of a 3-manifold and the Thurston norm on cohomology, to appear in Ann. Sci. École. Norm. Sup. (4).
- Alan W. Reid, Arithmeticity of knot complements, J. London Math. Soc. (2) 43 (1991), no. 1, 171–184. MR 1099096, DOI 10.1112/jlms/s2-43.1.171
- Józef Marcinkiewicz and Antoni Zygmund, Sur la dérivée seconde généralisée, Bull. Sém. Math. Univ. Wilno 2 (1939), 35–40 (French). MR 45
- Leo F. Epstein, A function related to the series for $e^{e^x}$, J. Math. Phys. Mass. Inst. Tech. 18 (1939), 153–173. MR 58, DOI 10.1002/sapm1939181153
- P. B. Shalen, Representations of 3-manifold groups, Handbook of geometric topology, Elsevier Press, to appear.
- L. E. Dickson, All integers except $23$ and $239$ are sums of eight cubes, Bull. Amer. Math. Soc. 45 (1939), 588–591. MR 28, DOI 10.1090/S0002-9904-1939-07041-9
- William P. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 3, 357–381. MR 648524, DOI 10.1090/S0273-0979-1982-15003-0
- Antoni Zygmund, Sur un théorèm de M. Fejér, Bull. Sém. Math. Univ. Wilno 2 (1939), 3–12 (French). MR 52
- J. Weeks, SnapPea, http://www.northnet.org/weeks/.
- Waterloo Maple Software, Maple 6, 2000.
Additional Information
- Danny Calegari
- Affiliation: Department of Mathematics, Harvard University, Cambridge Massachusetts 02138
- MR Author ID: 605373
- Email: dannyc@math.harvard.edu
- Nathan M. Dunfield
- Affiliation: Department of Mathematics, Harvard University, Cambridge Massachusetts 02138
- MR Author ID: 341957
- ORCID: 0000-0002-9152-6598
- Email: nathand@math.harvard.edu
- Received by editor(s): February 7, 2001
- Received by editor(s) in revised form: August 25, 2001
- Published electronically: February 25, 2002
- Additional Notes: Both authors were partially supported by the National Science Foundation.
- © Copyright 2002 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 354 (2002), 2955-2969
- MSC (2000): Primary 57M25, 57M50
- DOI: https://doi.org/10.1090/S0002-9947-02-02988-4
- MathSciNet review: 1895211