Skip to main content
SpringerLink
Log in

The Colored Jones Polynomial, the Chern–Simons Invariant, and the Reidemeister Torsion of a Twice–Iterated Torus Knot

  • Published:
Acta Mathematica Vietnamica Aims and scope Submit manuscript

Abstract

A generalization of the volume conjecture relates the asymptotic behavior of the colored Jones polynomial of a knot to the Chern–Simons invariant and the Reidemeister torsion of the knot complement associated with a representation of the fundamental group to the special linear group of degree two over complex numbers. If the knot is hyperbolic, the representation can be regarded as a deformation of the holonomy representation that determines the complete hyperbolic structure. In this article, we study a similar phenomenon when the knot is a twice-iterated torus knot. In this case, the asymptotic expansion of the colored Jones polynomial splits into sums, and each summand is related to the Chern–Simons invariant and the Reidemeister torsion associated with a representation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Chern, S.-S., Simons, J.: Characteristic forms and geometric invariants. Ann. Math. 99 (2), 48–69 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  2. Dimofte, T., Gukov, S.: Quantum field theory and the volume conjecture, interactions between hyperbolic geometry, quantum topology and number theory, contemporary mathematics, Vol. 541, pp 41–67. American Mathematical Society, Providence (2011)

  3. Dubois, J.: Non abelian twisted Reidemeister torsion for fibered knots. Canad. Math. Bull. 49 (1), 55–71 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  4. Dubois, J., Huynh, V., Yamaguchi, Y.: Non-abelian Reidemeister torsion for twist knots. J. Knot Theory Ramifications 18 (3), 303–341 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  5. Dubois, J., Kashaev, R.M.: On the asymptotic expansion of the colored jones polynomial for torus knots. Math. Ann. 339 (4), 757–782 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  6. Eisenbud, D., Neumann, W.: Three-dimensional link theory and invariants of plane curve singularities Annals of Mathematics Studies, Vol. 110. Princeton University Press, Princeton (1985)

  7. Fox, R.H.: Free differential calculus. I. Derivation in the free group ring. Ann. of Math. (2) 57, 547–560 (1953)

    Article  MathSciNet  Google Scholar 

  8. Gromov, M.: Volume and bounded cohomology. Inst. Hautes Études Sci. Publ. Math. 56, 5–99 (1982)

  9. Gukov, S.: Three-dimensional quantum gravity, Chern-Simons theory, and the A-polynomial. Comm. Math. Phys. 255 (3), 577–627 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  10. Gukov, S., Murakami, H.: SL(2;C) Chern-Simons theory and the asymptotic behavior of the colored Jones polynomial. In: Yui, N., Verrill, H., Doran, C.F. (eds.) Modular forms and string duality fields institute commission, Vol. 54, pp 261–278. American Mathematical Society and Fields Institute (2008)

  11. Heusener, M.: An orientation for the SU(2)-representation space of knot groups In: Proceedings of the pacific institute for the mathematical sciences workshop invariants of three- manifolds (Calgary, AB, 1999), Vol. 127, pp 175–197 (2003)

  12. Hikami, K., Murakami, H.: Colored Jones polynomials with polynomial growth. Commun. Contemp. Math. 10 (suppl. 1), 815–834 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  13. Hikami, K., Murakami, H.: Representations and the colored Jones polynomial of a torus knot, Chern-Simons gauge theory: 20 years after, AMS/IP Studies Advances in Mathematics, Vol. 50, pp 153–171. American Mathematical Society, Providence (2011)

  14. Hodgson, C.D., Kerckhoff, S.P.: Rigidity of hyperbolic cone-manifolds and hyperbolic Dehn surgery. J. Differ. Geom. 48 (1), 1–59 (1998)

    MATH  MathSciNet  Google Scholar 

  15. Jaco, W.H., Shalen, P.B.: Seifert fibered spaces in 3-manifolds. Mem. Am. Math. Soc. 21 (220), viii + 192 (1979)

    MathSciNet  Google Scholar 

  16. Johannson, K.: Homotopy equivalences of 3-manifolds with boundaries Lecture notes in mathematics, Vol. 761. Springer, Berlin (1979)

  17. Jones, V.F.R.: A polynomial invariant for knots via von Neumann algebras. Bull. Am. Math. Soc. (N.S.) 12 (1), 103–111 (1985)

    Article  MATH  Google Scholar 

  18. Kashaev, R.M.: A link invariant from quantum dilogarithm. Modern Phys. Lett. A 10 (19), 1409–1418 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  19. Kashaev, R.M.: The hyperbolic volume of knots from the quantum dilogarithm. Lett. Math. Phys. 39 (3), 269–275 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  20. Kashaev, R.M., Tirkkonen, O.: A proof of the volume conjecture on torus knots. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 269, no. Vopr. Kvant. Teor. Polya i Stat. Fiz. 16, pp.262–268, 370 (2000)

  21. Kauffman, L.H.: State models and the Jones polynomial. Topology 26 (3), 395–407 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  22. Kirillov, A.N., Reshetikhin, N.Y: Representations of the algebra Uq(sl(2)); q-orthogonal polynomials and invariants of links, Infinite-dimensional Lie algebras and groups (Luminy- Marseille) Advanced series mathematical physics, Vol. 7, pp 285–339. World Science Publishing, Teaneck, NJ (1989)

  23. Kirk, P., Klassen, E.: Chern-Simons invariants of 3-manifolds decomposed along tori and the circle bundle over the representation space of T2. Comm. Math. Phys. 153 (3), 521–557 (1993)

  24. Kirk, P., Livingston, C.: Twisted Alexander invariants, Reidemeister torsion, and Casson- Gordon invariants. Topology 38 (3), 635–661 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  25. Kitano, T.: Twisted Alexander polynomial and Reidemeister torsion. Pacific J. Math. 174 (2), 431–442 (1996)

    MATH  MathSciNet  Google Scholar 

  26. Klassen, E.P.: Representations of knot groups in SU(2). Trans. Am. Math. Soc. 326 (2), 795–828 (1991)

    MATH  MathSciNet  Google Scholar 

  27. Le, T.T.Q.: Varieties of representations and their cohomology-jump subvarieties for knot groups. Russ. Acad. Sci., Sb., Math. 78 (1), 187–209 (1993). (English. Russian original)

    Google Scholar 

  28. Lickorish, W.B.R.: An introduction to knot theory Graduate texts in mathematics, Vol. 175. Springer-Verlag, New York (1997)

  29. Marsden, J.E., Hoffman, M.J.: Basic complex analysis. W. H. Freeman and Company, New York (1987)

    MATH  Google Scholar 

  30. Masbaum, G., Vogel, P.: 3-valent graphs and the Kauffman bracket. Pacific J. Math. 164 (2), 361–381 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  31. Milnor, J.: A duality theorem for Reidemeister torsion. Ann. Math. (2) 76, 137–147 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  32. Milnor, J.: Whitehead torsion. Bull. Am. Math. Soc. 72, 358–426 (1966)

    Article  MATH  MathSciNet  Google Scholar 

  33. Milnor, J.: Infinite cyclic coverings In: Conference on the Topology of Manifolds(Michigan State University, E. Lansing, Michigan, 1967), pp 115–133. Prindle, Weber & Schmidt, Boston (1968)

    Google Scholar 

  34. Milnor, J.: Hyperbolic geometry: the 1st 150 years. Bull. Am. Math. Soc. (N.S.) 6 (1), 9–24 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  35. Morton, H.R.: The coloured Jones function and Alexander polynomial for torus knots. Math. Proc. Cambridge Philos. Soc. 117 (1), 129–135 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  36. Murakami, H.: Various generalizations of the volume conjecture The interaction of analysis and geometry on contemporary mathematics, Vol. 424, pp 165–186. American Mathematical Society, Providence (2007)

  37. Murakami, H.: A version of the volume conjecture. Adv. Math. 211 (2), 678–683 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  38. Murakami, H.: An introduction to the volume conjecture and its generalizations. Acta Math. Vietnam. 33 (3), 219–253 (2008)

    MATH  MathSciNet  Google Scholar 

  39. Murakami, H.: The coloured Jones polynomial, the Chern-Simons invariant, and the Reidemeister torsion of the figure-eight knot. J. Topol. 6 (1), 193–216 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  40. Murakami, H., Murakami, J.: The colored Jones polynomials and the simplicial volume of a knot. Acta Math. 186 (1), 85–104 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  41. Murakami, H., Yokota, Y.: The colored Jones polynomials of the figure-eight knot and its Dehn surgery spaces. J. Reine Angew. Math. 607, 47–68 (2007)

    MATH  MathSciNet  Google Scholar 

  42. Neumann, W.D., Zagier, D.: Volumes of hyperbolic three-manifolds. Topology 24 (3), 307–332 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  43. Porti, J.: Torsion de Reidemeister pour les variétés hyperboliques. Mem. Am. Math. Soc. 128 (612), x + 139 (1997)

    MathSciNet  Google Scholar 

  44. Riley, R.: A quadratic parabolic group. Math. Proc. Cambridge Philos. Soc. 77, 281–288 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  45. Riley, R.: Nonabelian representations of 2-bridge knot groups. Quart. J. Math. Oxford Ser. (2) 35 (138), 191–208 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  46. Rosso, M., Jones, V.: On the invariants of torus knots derived from quantum groups. J. Knot Theory Ramifications 2 (1), 97–112 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  47. Thurston, W.P.: The geometry and topology of three-manifolds. Electronic version 1.1 (2002). http://www.msri.org/publications/books/gt3m/

  48. Turaev, V.: Reidemeister torsion in knot theory. Uspekhi Mat. Nauk 41 (1(247)), 240 (1986)

    MathSciNet  Google Scholar 

  49. Turaev, V.: Introduction to combinatorial torsions Lectures in Mathematics ETH Zürich. Notes taken by Felix Schlenk. Birkhäuser Verlag, Basel (2001)

    Google Scholar 

  50. van der Veen, R.: A cabling formula for the colored Jones polynomial. arXiv: http://arxiv.org/abs/0807.2679 (2008)

  51. Waldhausen, F.: Algebraic K-theory of generalized free products. I, II. Ann. of Math. (2) 108 (1), 135–204 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  52. Wenzl, H.: On sequences of projections. C. R. Math. Rep. Acad. Sci. Canada 9 (1), 5–9 (1987)

    MATH  MathSciNet  Google Scholar 

  53. Wolfram Research, Inc.: Mathematica 9 (2013)

  54. Yamaguchi, Y.: A relationship between the non-acyclic Reidemeister torsion and a zero of the acyclic Reidemeister torsion. Ann. Inst. Fourier (Grenoble) 58 (1), 337–362 (2008)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

This article is prepared for the proceedings of the conference “The Quantum Topology and Hyperbolic Geometry” in Nha Trang, Vietnam, 13–17 May, 2013. I would like to thank the organizers for their hospitality.

Part of this work was done when the author was visiting the Max-Planck Institute for Mathematics, Université Paris Diderot, and the University of Amsterdam. The author thanks Christian Blanchet, Roland van der Veen, Jinseok Cho, and Satoshi Nawata for helpful discussions.

This work was supported by JSPS KAKENHI Grant Numbers 23340115, 24654041.

Author information

Authors and Affiliations

  1. Graduate School of Information Sciences, Tohoku University, Aramaki-aza-Aoba 6-3-09, Aoba-ku, Sendai, 980-8579, Japan

    Hitoshi Murakami

Authors
  1. Hitoshi Murakami
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hitoshi Murakami.

Electronic Supplementary Material

Below is the link to the electronic supplementary material.

ESM 1

(.STY 4.62 KB)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Murakami, H. The Colored Jones Polynomial, the Chern–Simons Invariant, and the Reidemeister Torsion of a Twice–Iterated Torus Knot. Acta Math Vietnam 39, 649–710 (2014). https://doi.org/10.1007/s40306-014-0084-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40306-014-0084-x

Keywords

Mathematics Subject Classification (2010)

Search

Navigation