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New formulas for the character varieties of two-bridge links

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Abstract

We consider some families of two-bridge links, including the links \(\mathbf{b}(2 p, 3)\), the twisted Whitehead links and the double twist links, and calculate their character varieties. Then we give simple geometrical descriptions of such varieties, and determine the number of their irreducible components. Our paper relates to the nice work of Hilden, Lozano and Montesinos on the character variety of a class of 2-bridge links, but uses very different methods based on the concept of palindrome presentations of link groups. The obtained formulas for the above character varieties are new, easily programmable, and more simple in many cases than the equations known in the literature.

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References

  1. Cavicchioli, A., Repovš, D., Spaggiari, F.: Palindrome presentations of rational knots. J. Knot Theory Ramif. 18(3), 343–361 (2009)

    Article  MathSciNet  Google Scholar 

  2. Cavicchioli, A., Spaggiari, F.: The character variety of some classes of rational knots. J. Knot Theory Ramif. 28(9), 1950055, 15 (2019)

  3. Culler, M., Shalen, P.B.: Varieties of group representations and splitting of 3-manifolds. Ann. Math. 117(2), 109–146 (1983)

    Article  MathSciNet  Google Scholar 

  4. Gonzáles Acuña, F., Montesinos-Amilibia, J.M.: On the character variety of group representations in\({\rm SL}(2, {C\mathit{}) and {\rm PSL}(2, {C}})\). Math. Z. 214(1), 627–652 (1993)

  5. Ham-J, J.Y., Lee, J., Mednykh, A., Rasskazov, A.: On the volume and Chern-Simons invariant for 2-bridge knot orbifolds. J. Knot Theory Ramif. 26(12), 1750082 (2017)

    Article  MathSciNet  Google Scholar 

  6. Harada, S.: Canonical components of character varieties of arithmetic two-bridge link complements. arXiv:1112.3441

  7. Hilden, H.M., Lozano, M.T., Montesinos-Amilibia, J.M.: On the character variety of group representations of a 2-bridge link p/3 into PSL\((2, {\mathbb{C}})\). Bol. Soc. Mat. Mexicana 37(2), 241–262 (1992)

  8. Hilden, H.M., Lozano, M.T., Montesinos-Amilibia, J.M.: On the character variety of tunnel number 1 knots. J. London Math. Soc. 62(3), 938–950 (2000)

  9. Hsiao, H.: On factorization of Chebyshev’s polynomials of the first kind. Bull. Inst. Math. Acad. Sinica 12(1), 89–94 (1984)

  10. Heusener, M., Porti, J.: The variety of characters in\({\rm PSL}_2({\mathbb{C}})\). Bol. Soc. Mat. Mexicana 10(3), 221–237 (2004)

  11. Kawauchi, K.: A Survey of Knot Theory. Birkhäuser Verlag, Basel (1996)

    MATH  Google Scholar 

  12. Landes, E.: Identifying the canonical component for the Whitehead link. Mat. Res. Lett. 18, 715–721 (2011)

    Article  MathSciNet  Google Scholar 

  13. Le, T.T.Q., Tran, A.T.: The Kauffman bracket skein module of two-bridge links. Proc. Am. Math. Soc. 142, 1045–1056 (2014)

    Article  MathSciNet  Google Scholar 

  14. Lubotzky, A., Magid, A.: Varieties of representations of finitely generated groups. Mem. Am Math. Soc. 58,(1985)

  15. Macasieb, M.L., Petersen, K.L., van Luijk, R.M.: On character varieties of two-bridge knot groups. Proc. London Math. Soc 103(3), 473–507 (2011)

    Article  MathSciNet  Google Scholar 

  16. Mednykh, A.D.: Volumes of two-bridge cone manifolds in spaces of constant curvature. Transf. Groups 26(2), 601–629 (2021)

    Article  MathSciNet  Google Scholar 

  17. Molnár, E., Szirmai, J., Vesnin, A.: Projective metric realizations of cone-manifolds with singularities along 2-bridge knots and links. J. Geom. 95(1–2), 91–133 (2009)

  18. Munoz, V.: The\({\rm SL}(2, {\mathbb{C}})\)character varieties of torus knots. Rev. Mat. Complut. 22(2), 489–497 (2009)

  19. Nagasato, F., Tran, A.T.: Presentations of character varieties of 2-bridge knots using Chebyshev polynomials. arXiv:1301.0138 (2013)

  20. Petersen, K.L., Tran, A.T.: Character varieties of double twist links. Algebr. Geometr. Topol. 15(6), 3569–3598 (2015)

    Article  MathSciNet  Google Scholar 

  21. Qazaqzeh, K.: The character variety of a class of rational links. Turkish J. Math. 36, 41–46 (2012)

    MathSciNet  MATH  Google Scholar 

  22. Riley, R.: Algebra for Heckoid groups. Trans. Am. Math. Soc. 334(1), 389–409 (1992)

    Article  MathSciNet  Google Scholar 

  23. Rolfsen, D.: Knots and Links, Math. Lect. Series 7. Publish or Perish Inc., Berkeley (1976)

  24. Thurston, W.: Three-dimensional Geometry and Topology, Princeton Math. Ser. 35 (S. Levy, ed.). Princeton Univ. Press, Princeton, NJ (1997)

  25. Tran, A.T.: Character varieties of\((- 2, 2 m + 1, 2 n)\)-pretzel links and twisted Whitehead links. J. Knot Theory Ramifications 25(2), 1650007, 16 (2016)

  26. Tran, A.T.: The universal character ring of the\((- 2, 2 m + 1, 2 n)\)-pretzel link. Internat. J. Math. 24(8), 1350063, 13 (2013)

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Acknowledgements

Work performed under the auspices of the scientific group G.N.S.A.G.A. of the C.N.R (National Research Council) of Italy and partially supported by the MIUR (Ministero dell’ Istruzione, dell’ Universitá e della Ricerca ) of Italy within the project Strutture Geometriche, Combinatoria e loro Applicazioni, and by a research grant FAR 2020 of the University of Modena and Reggio Emilia. The authors would like to thank the anonymous referee for his/her useful suggestions, which improved the final version of the paper.

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  1. Alberto Cavicchioli, Dipartimento di Scienze Fisiche, Informatiche e Matematiche, Universitá di Modena e Reggio Emilia, Via Campi 213/B, Modena, 41125, Italy

    Alberto Cavicchioli & Fulvia Spaggiari

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  1. Alberto Cavicchioli
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  2. Fulvia Spaggiari
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Correspondence to Alberto Cavicchioli.

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Cavicchioli, A., Spaggiari, F. New formulas for the character varieties of two-bridge links. J. Geom. 112, 43 (2021). https://doi.org/10.1007/s00022-021-00608-0

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