Acta Mathematica Vietnamica

, Volume 39, Issue 4, pp 649–710 | Cite as

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

  • Hitoshi MurakamiEmail author


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.


Knot Volume conjecture Colored Jones polynomial Chern-Simons invariant Reidemeister torsion Iterated torus knot 

Mathematics Subject Classification (2010)

Primary 57M27 Secondary 57M25 57M50 58J28 



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.

Supplementary material

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ESM 1 (.STY 4.62 KB)


  1. 1.
    Chern, S.-S., Simons, J.: Characteristic forms and geometric invariants. Ann. Math. 99 (2), 48–69 (1974)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 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)Google Scholar
  3. 3.
    Dubois, J.: Non abelian twisted Reidemeister torsion for fibered knots. Canad. Math. Bull. 49 (1), 55–71 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Dubois, J., Huynh, V., Yamaguchi, Y.: Non-abelian Reidemeister torsion for twist knots. J. Knot Theory Ramifications 18 (3), 303–341 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 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)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 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)Google Scholar
  7. 7.
    Fox, R.H.: Free differential calculus. I. Derivation in the free group ring. Ann. of Math. (2) 57, 547–560 (1953)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Gromov, M.: Volume and bounded cohomology. Inst. Hautes Études Sci. Publ. Math. 56, 5–99 (1982)Google Scholar
  9. 9.
    Gukov, S.: Three-dimensional quantum gravity, Chern-Simons theory, and the A-polynomial. Comm. Math. Phys. 255 (3), 577–627 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 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)Google Scholar
  11. 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)Google Scholar
  12. 12.
    Hikami, K., Murakami, H.: Colored Jones polynomials with polynomial growth. Commun. Contemp. Math. 10 (suppl. 1), 815–834 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 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)Google Scholar
  14. 14.
    Hodgson, C.D., Kerckhoff, S.P.: Rigidity of hyperbolic cone-manifolds and hyperbolic Dehn surgery. J. Differ. Geom. 48 (1), 1–59 (1998)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Jaco, W.H., Shalen, P.B.: Seifert fibered spaces in 3-manifolds. Mem. Am. Math. Soc. 21 (220), viii + 192 (1979)MathSciNetGoogle Scholar
  16. 16.
    Johannson, K.: Homotopy equivalences of 3-manifolds with boundaries Lecture notes in mathematics, Vol. 761. Springer, Berlin (1979)Google Scholar
  17. 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)CrossRefzbMATHGoogle Scholar
  18. 18.
    Kashaev, R.M.: A link invariant from quantum dilogarithm. Modern Phys. Lett. A 10 (19), 1409–1418 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Kashaev, R.M.: The hyperbolic volume of knots from the quantum dilogarithm. Lett. Math. Phys. 39 (3), 269–275 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 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)Google Scholar
  21. 21.
    Kauffman, L.H.: State models and the Jones polynomial. Topology 26 (3), 395–407 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 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)Google Scholar
  23. 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)Google Scholar
  24. 24.
    Kirk, P., Livingston, C.: Twisted Alexander invariants, Reidemeister torsion, and Casson- Gordon invariants. Topology 38 (3), 635–661 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Kitano, T.: Twisted Alexander polynomial and Reidemeister torsion. Pacific J. Math. 174 (2), 431–442 (1996)zbMATHMathSciNetGoogle Scholar
  26. 26.
    Klassen, E.P.: Representations of knot groups in SU(2). Trans. Am. Math. Soc. 326 (2), 795–828 (1991)zbMATHMathSciNetGoogle Scholar
  27. 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. 28.
    Lickorish, W.B.R.: An introduction to knot theory Graduate texts in mathematics, Vol. 175. Springer-Verlag, New York (1997)Google Scholar
  29. 29.
    Marsden, J.E., Hoffman, M.J.: Basic complex analysis. W. H. Freeman and Company, New York (1987)zbMATHGoogle Scholar
  30. 30.
    Masbaum, G., Vogel, P.: 3-valent graphs and the Kauffman bracket. Pacific J. Math. 164 (2), 361–381 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Milnor, J.: A duality theorem for Reidemeister torsion. Ann. Math. (2) 76, 137–147 (1962)CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Milnor, J.: Whitehead torsion. Bull. Am. Math. Soc. 72, 358–426 (1966)CrossRefzbMATHMathSciNetGoogle Scholar
  33. 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. 34.
    Milnor, J.: Hyperbolic geometry: the 1st 150 years. Bull. Am. Math. Soc. (N.S.) 6 (1), 9–24 (1982)CrossRefzbMATHMathSciNetGoogle Scholar
  35. 35.
    Morton, H.R.: The coloured Jones function and Alexander polynomial for torus knots. Math. Proc. Cambridge Philos. Soc. 117 (1), 129–135 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  36. 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)Google Scholar
  37. 37.
    Murakami, H.: A version of the volume conjecture. Adv. Math. 211 (2), 678–683 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  38. 38.
    Murakami, H.: An introduction to the volume conjecture and its generalizations. Acta Math. Vietnam. 33 (3), 219–253 (2008)zbMATHMathSciNetGoogle Scholar
  39. 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)CrossRefzbMATHMathSciNetGoogle Scholar
  40. 40.
    Murakami, H., Murakami, J.: The colored Jones polynomials and the simplicial volume of a knot. Acta Math. 186 (1), 85–104 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  41. 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)zbMATHMathSciNetGoogle Scholar
  42. 42.
    Neumann, W.D., Zagier, D.: Volumes of hyperbolic three-manifolds. Topology 24 (3), 307–332 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  43. 43.
    Porti, J.: Torsion de Reidemeister pour les variétés hyperboliques. Mem. Am. Math. Soc. 128 (612), x + 139 (1997)MathSciNetGoogle Scholar
  44. 44.
    Riley, R.: A quadratic parabolic group. Math. Proc. Cambridge Philos. Soc. 77, 281–288 (1975)CrossRefzbMATHMathSciNetGoogle Scholar
  45. 45.
    Riley, R.: Nonabelian representations of 2-bridge knot groups. Quart. J. Math. Oxford Ser. (2) 35 (138), 191–208 (1984)CrossRefzbMATHMathSciNetGoogle Scholar
  46. 46.
    Rosso, M., Jones, V.: On the invariants of torus knots derived from quantum groups. J. Knot Theory Ramifications 2 (1), 97–112 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  47. 47.
    Thurston, W.P.: The geometry and topology of three-manifolds. Electronic version 1.1 (2002).
  48. 48.
    Turaev, V.: Reidemeister torsion in knot theory. Uspekhi Mat. Nauk 41 (1(247)), 240 (1986)MathSciNetGoogle Scholar
  49. 49.
    Turaev, V.: Introduction to combinatorial torsions Lectures in Mathematics ETH Zürich. Notes taken by Felix Schlenk. Birkhäuser Verlag, Basel (2001)Google Scholar
  50. 50.
    van der Veen, R.: A cabling formula for the colored Jones polynomial. arXiv: (2008)
  51. 51.
    Waldhausen, F.: Algebraic K-theory of generalized free products. I, II. Ann. of Math. (2) 108 (1), 135–204 (1978)CrossRefzbMATHMathSciNetGoogle Scholar
  52. 52.
    Wenzl, H.: On sequences of projections. C. R. Math. Rep. Acad. Sci. Canada 9 (1), 5–9 (1987)zbMATHMathSciNetGoogle Scholar
  53. 53.
    Wolfram Research, Inc.: Mathematica 9 (2013)Google Scholar
  54. 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)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2014

Authors and Affiliations

  1. 1.Graduate School of Information SciencesTohoku UniversitySendaiJapan

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