Hostname: page-component-8448b6f56d-xtgtn Total loading time: 0 Render date: 2024-04-23T16:48:32.645Z Has data issue: false hasContentIssue false
  • English
  • Français

Invariant character varieties of hyperbolic knots with symmetries

Published online by Cambridge University Press:  22 May 2017

LUISA PAOLUZZI  and
JOAN PORTI
LUISA PAOLUZZI
Affiliation:
Aix-Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, 13453 Marseille, France. e-mail: luisa.paoluzzi@univ-amu.fr
JOAN PORTI
Affiliation:
Departament de Matemàtiques, Universitat Autònoma de Barcelona and BGSMath 08193 Bellaterra, Spain. e-mail: porti@mat.uab.cat
Get access Rights & Permissions [Opens in a new window]

Abstract

We study character varieties of symmetric knots and their reductions mod p. We observe that the varieties present a different behaviour according to whether the knots admit a free or periodic symmetry.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 165 , Issue 2 , September 2018 , pp. 193 - 208
Copyright
Copyright © Cambridge Philosophical Society 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

†

Partially supported by ANR project 12-BS01-0003-01.

‡

Partially supported by the Spanish Mineco through grant MTM2015-66165-P.

References

REFERENCES

[1] Boileau, M. and Porti, J. Geometrisation of 3-orbifolds of cyclic type. Astérisque 272 (2001), 208. Appendix A by Heusener, M. and Porti, J..Google Scholar
[2] Boileau, M. and Zimmermann, B. Symmetries of nonelliptic Montesinos links. Math. Ann. 277 (1987), 563–584.Google Scholar
[3] Brown, K. S. Cohomology of groups. Graduate Texts in Math. 87 (Springer-Verlag, New York, 1994). Corrected reprint of the 1982 original.Google Scholar
[4] Cox, D., Little, J. and O'Shea, D. An introduction to computational algebraic geometry and commutative algebra. Undergraduate Texts in Mathematics. (Springer, New York, 2007), xvi+551.Google Scholar
[5] Culler, M. and Shalen, P. B. Varieties of group representations and splittings of 3-manifolds. Ann. of Math. 117 (1983), 109–146.Google Scholar
[6] González-Acuña, F. and Montesinos–Amilibia, J. M. On the character variety of group representations in SL(2, C) and PSL(2, C). Math. Z. 214 (1993), 627–652.Google Scholar
[7] Harada, S. Canonical components of character varieties of arithmetic two bridge link complements. arXiv:1112.3441 (2011).Google Scholar
[8] Hatcher, A. Notes on basic 3-manifold topology. Available at https://www.math.cornell.edu/hatcher/3M/3Mdownloads.html.Google Scholar
[9] Hilden, H. M., Lozano, M. T. and Montesinos–Amilibia, J. M. On the character variety of periodic knots and links. Math. Proc. Camb. Phil. Soc. 129 (2000), 477–490.Google Scholar
[10] Jaco, W. H. and Shalen, P. B. Seifert fibred spaces in 3-manifolds. Mem. Amer. Math. Soc. 220 (1979), viii+192.Google Scholar
[11] Johannson, K. Homotopy equivalence of 3-manifolds with boundary. Lecture Notes in Math. 761 (Springer-Verlag, Berlin, 1979).Google Scholar
[12] Li, Y. 1-dominations between Montesinos knots with at least 6 rational tangles. Topology Appl. 157 (2010), 1033–1043.Google Scholar
[13] Lubotzky, A. and Magid, A. R. Varieties of representations of finitely generated groups. Mem. Amer. Math. Soc. 336 (1985), xi+117.Google Scholar
[14] Morgan, J. W. and Bass, H. The Smith conjecture. Pure and Appl. Math. 112. (Academic Press, Inc., Orlando, FL, 1984). Papers presented at the symposium held at Columbia University, New York, 1979.Google Scholar
[15] Paoluzzi, L. and Porti, J. Non-standard components of the character variety for a family of Montesinos knots. Proc. Lond. Math. Soc. 107 (2013), 655–679.Google Scholar
[16] Porti, J. Hyperbolic polygons of minimal perimeter with given angles. Geom. Dedicata 156 (2012), 165–170.Google Scholar
[17] Porti, J. Regenerating hyperbolic cone 3-manifolds from dimension 2. Ann. Inst. Fourier (Grenoble) 63 (2013), 1971–2015.Google Scholar
[18] Rong, Y. Degree one maps between geometric 3-manifolds. Trans. Amer. Math. Soc. 332 (1992), 411–436.Google Scholar
[19] Thurston, W. P. The Geometry and Topology of Three-Manifolds (Electronic version from 2002 available at http://www.msri.org/publications/books/gt3m/) (1980).Google Scholar
[20] Weil, A. Remarks on the cohomology of groups. Ann. of Math. 80 (1964), 149–157.Google Scholar