DOI QR코드

DOI QR Code

OPTIMISTIC LIMITS OF THE COLORED JONES POLYNOMIALS

  • Cho, Jinseok ;
  • Murakami, Jun
  • Received : 2012.10.15
  • Published : 2013.05.01

Abstract

We show that the optimistic limits of the colored Jones polynomials of the hyperbolic knots coincide with the optimistic limits of the Kashaev invariants modulo $4{\pi}^2$.

Keywords

volume conjecture;colored Jones polynomial;optimistic limit;Kashaev invariant

References

  1. J. Cho, Yokota theory, the invariant trace fields of hyperbolic knots and the Borel regulator map, http://arxiv.org/abs/1005.3094, 2010.
  2. J. Cho and J. Murakami, The complex volumes of twist knots via colored Jones polynomials, J. Knot Theory Ramifications 19 (2010), no. 11, 1401-1421. https://doi.org/10.1142/S0218216510008443
  3. J. Cho, J. Murakami, and Y. Yokota, The complex volumes of twist knots, Proc. Amer. Math. Soc. 137 (2009), no. 10, 3533-3541. https://doi.org/10.1090/S0002-9939-09-09906-7
  4. S. Francaviglia, Hyperbolic volume of representations of fundamental groups of cusped 3-manifolds, Int. Math. Res. Not. 2004 (2004), no. 9, 425-459. https://doi.org/10.1155/S1073792804131619
  5. R. M. Kashaev, The hyperbolic volume of knots from the quantum dilogarithm, Lett. Math. Phys. 39 (1997), no. 3, 269-275. https://doi.org/10.1023/A:1007364912784
  6. L. Lewin, Structural Properties of Polylogarithms, Volume 37 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1991.
  7. R. Meyerhoff, Density of the Chern-Simons invariant for hyperbolic 3-manifolds, In Low-dimensional topology and Kleinian groups (Coventry/Durham, 1984), Volume 112 of London Math. Soc. Lecture Note Ser., pages 217-239. Cambridge Univ. Press, Cambridge, 1986.
  8. H. Murakami, The asymptotic behavior of the colored Jones function of a knot and its volume, Proceedings of 'Art of Low Dimensional Topology VI', edited by T. Kohno, January, 2000.
  9. H. Murakami, Optimistic calculations about the Witten-Reshetikhin-Turaev invariants of closed three-manifolds obtained from the figure-eight knot by integral Dehn surgeries, Surikaisekikenkyusho Kokyuroku, (1172):70-79, 2000, Recent progress towards the volume conjecture (Japanese) (Kyoto, 2000).
  10. H. Murakami, Kashaev's invariant and the volume of a hyperbolic knot after Y. Yokota, In Physics and combinatorics 1999 (Nagoya), pages 244-272, World Sci. Publ., River Edge, NJ, 2001.
  11. H.Murakami and J. Murakami, The colored Jones polynomials and the simplicial volume of a knot, Acta Math. 186 (2001), no. 1, 85-104. https://doi.org/10.1007/BF02392716
  12. H. Murakami, J. Murakami, M. Okamoto, T. Takata, and Y. Yokota, Kashaev's conjec- ture and the Chern-Simons invariants of knots and links, Experiment. Math. 11 (2002), no. 3, 427-435. https://doi.org/10.1080/10586458.2002.10504485
  13. K. Ohnuki, The colored Jones polynomials of 2-bridge link and hyperbolicity equations of its complements, J. Knot Theory Ramifications 14 (2005), no. 6, 751-771. https://doi.org/10.1142/S021821650500407X
  14. D. Thurston, Hyperbolic volume and the Jones polynomial, Lecture note at "Invariants des noeuds et de varietes de dimension 3", available at http://www.math.columbia.edu/-dpt/speaking/Grenoble.pdf, June 1999.
  15. W. Thurston, The geometry and topology of three-manifolds, Lecture Note. available at http://www.msri.org/publications/books/gt3m/.
  16. S. Tillmann, Degenerations of ideal hyperbolic triangulations, http://arxiv.org/abs/math/0508295.
  17. Y. Yokota, On the volume conjecture for hyperbolic knots, http://arxiv.org/abs/math/0009165.
  18. Y. Yokota, On the complex volume of hyperbolic knots, J. Knot Theory Ramifications 20 (2011), no. 7, 955-976. https://doi.org/10.1142/S021821651100908X
  19. C. K. Zickert, The volume and Chern-Simons invariant of a representation, Duke Math. J. 150 (2009), no. 3, 489-532. https://doi.org/10.1215/00127094-2009-058

Cited by

  1. OPTIMISTIC LIMITS OF THE COLORED JONES POLYNOMIALS AND THE COMPLEX VOLUMES OF HYPERBOLIC LINKS vol.100, pp.03, 2016, https://doi.org/10.1017/S144678871600001X
  2. Reidemeister transformations of the potential function and the solution 2017, https://doi.org/10.1142/S0218216517500791
  3. Optimistic limits of Kashaev invariants and complex volumes of hyperbolic links vol.23, pp.09, 2014, https://doi.org/10.1142/S0218216514500497
  4. Octahedral developing of knot complement I: Pseudo-hyperbolic structure vol.197, pp.1, 2018, https://doi.org/10.1007/s10711-018-0323-8

Acknowledgement

Supported by : Korea Research Foundation