Generalized volume conjecture and the A-polynomials: The Neumann-Zagier potential function as a classical limit of the partition function

Journal of Geometry and Physics

Volumn 57, Issue 9, 2007, Pages 1895-1940

  Hikami, Kazuhiro ^a  

^a UNIVERSITY OF TOKYO   (Japan)

Author keywords

A polynomial; Chern Simons theory; Hyperbolic manifold; Jones polynomial; Neumann Zagier function; Quantum dilogarithm function; Quantum invariant

Indexed keywords

EID: 34248578237     PISSN: 03930440     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.geomphys.2007.03.008     Document Type: Article

References (64)

1. Barnes E.W. Theory of the double gamma functions. Philos. Trans. R. Soc. A 196 (1901) 265-388
2. Baseilhac S., and Benedetti R. Quantum hyperbolic invariants of 3-manifolds with PSL(2,C) characters. Topology 43 (2004) 1373-1423
3. Baseilhac S., and Benedetti R. Classical and quantum dilogarithmic invariants of flat PSL(2,C)-bundles over 3-manifolds. Geom. Topol. 9 (2005) 493-569
4. Bazhanov V.V., and Reshetikhin N. Remarks on the quantum dilogarithm. J. Phys. A: Math. Gen. 28 (1995) 2217-2226
5. Benedetti R., and Petronio C. Lectures on Hyperbolic Geometry (1992), Universitext, Springer-Verlag, Berlin
6. 1. Kobe J. Math. 3 (1986) 117-147
7. Bonahon F., and Liu X. Representation of the quantum Teichmüller space and invariants of surface diffeomorphism (2004). math.GT/0407086
8. q (s l (2, R)). Comm. Math. Phys. 240 (2003) 171-196
9. Callahan P.J., Dean J.C., and Weeks J.R. The simplest hyperbolic knots. J. Knot Theory Ramifications 8 (1999) 279-297
10. Chekhov L., and Fock V.V. Quantum Teichmüller space. Theoret. Math. Phys. 120 (1999) 1245-1259
11. Chekhov L.O., and Penner R.C. Introduction to quantum Thurston theory. Russian Math. Surveys 58 (2003) 1141-1183
12. Cooper D., Culler M., Gillet H., Long D.D., and Shalen P.B. Plane curves associated to character varieties of 3-manifolds. Invent. Math. 118 (1994) 47-84
13. Cooper D., and Long D.D. Representation theory and the A-polynomial of a knot - Knot theory and its applications. Chaos Solitons Fractals 9 (1998) 749-763
14. Coulson D., Goodman O.A., Hodgson C.D., and Neumann W.D. Computing arithmetic invariants of 3-manifolds. Experiment. Math. 9 (2000) 127-152
15. Dupont J.L., and Zickert C.K. A dilogarithmic formula for the Cheeger-Chern-Simons class. Geom. Topol. 10 (2006) 1347-1372
16. Faddeev L.D. Modular double of quantum group. In: Dito G., and Sternheimer D. (Eds). Conference Mosh Flato 1999 - Vol. I. Quantization, Deformations, and Symmetries. Math. Phys. Stud. vol. 21 (2000), Kluwer, Dordrecht 149-156
17. Faddeev L.D., and Kashaev R.M. Quantum dilogarithm. Modern Phys. Lett. A 9 (1994) 427-434
18. Faddeev L.D., Kashaev R.M., and Volkov A.Yu. Strongly coupled quantum discrete Liouville theory I: Algebraic approach and duality. Comm. Math. Phys. 219 (2001) 199-219
19. Floyd W., and Hatcher A. Incompressible surfaces in punctured torus bundles. Topology Appl. 13 (1982) 263-282
20. Francaviglia S. Algebraic and geometric solutions of hyperbolicity equations. Topology Appl. 145 (2004) 91-118
21. Garoufalidis S., and Le T.T.Q. The colored Jones function is q-holonomic. Geom. Topol. 9 (2005) 1253-1293
22. Gukov S. Three-dimensional quantum gravity, Chern-Simons theory, and the A-polynomial. Comm. Math. Phys. 255 (2005) 577-627
23. Hikami K. Asymptotics of the colored Jones polynomial and the A-polynomial. Nuclear Phys. B (2007) 10.1016/j.nuclphysb.2007.03.022
24. Hikami K. Hyperbolic structure arising from a knot invariant. Internat. J. Modern Phys. A 16 (2001) 3309-3333
25. Hikami K. Hyperbolicity of partition function and quantum gravity. Nuclear Phys. B 616 (2001) 537-548
26. Hikami K. Hyperbolic structure arising from a knot invariant II. Completeness, Internat. J. Modern Phys. B 16 (2002) 1963-1970
27. Hikami K. Difference equation of the colored Jones polynomial for the torus knot. Int. J. Math. 15 (2004) 959-965
28. Hoste J., and Thistlethwaite M.B. Knotscape (1999). http://www.math.utk.edu/~morwen/
29. Jones V.F.R. A polynomial invariant for knots via von Neumann algebras. Bull. Amer. Math. Soc. 12 (1985) 103-111
30. Kashaev R.M. Quantum dilogarithm as a 6 j-symbol. Modern Phys. Lett. A 9 (1994) 3757-3768
31. Kashaev R.M. An invariant of triangulated links from quantum dilogarithm. POMI 224 (1995) 208-214
32. Kashaev R.M. A link invariant from quantum dilogarithm. Modern Phys. Lett. A 10 (1995) 1409-1418
33. Kashaev R.M. The hyperbolic volume of knots from quantum dilogarithm. Lett. Math. Phys. 39 (1997) 269-275
34. Kashaev R.M. On the spectrum of Dehn twists in quantum Teichmüller theory. In: Kirillov A.N., and Liskova N. (Eds). Physics and Combinatorics - Proceedings of the Nagoya 2000 International Workshop (2001), World Scientific, Singapore 63-81
35. q (s l (2, R)), the modular double, and the multiparticle q-deformed Toda chains. Comm. Math. Phys. 225 (2002) 573-609
36. Kirillov A.N. Dilogarithm identities. Progr. Theoret. Phys. Suppl. 118 (1995) 61-142
37. q (S L (2)), q-orthogonal polynomials and invariants of links. In: Kac V.G. (Ed). Infinite Dimensional Lie Algebra and Groups (1989), World Scientific, Singapore 285-339
38. Kurokawa N. Multiple sine functions and Selberg zeta functions. Proc. Japan Acad. Ser. A Math. Sci. 67 (1991) 61-64
39. Le T.T.Q. The colored Jones polynomial and the A-polynomial of knots. Adv. Math. 207 (2006) 782-804
40. Lewin L. Polylogarithms and Associated Functions (1981), North-Holland, New York
41. Meyerhoff R. Hyperbolic 3-manifolds with equal volumes but different Chern-Simons invariants. In: Epstein D.B.A. (Ed). Low-Dimensional Topology and Kleinian Groups. London Math. Soc. Lecture Notes vol. 112 (1986), Cambridge Univ. Press, Cambridge 209-215
42. Milnor J. Hyperbolic geometry: The first 150 years. Bull. Amer. Math. Soc. 6 (1982) 9-24
43. Murakami H. Some limits of the colored Jones polynomials of the figure-eight knot. Kyungpook Math. J. 44 (2004) 369-383
44. Murakami H. A version of the volume conjecture (2006). math.GT/0603217
45. Murakami H., and Murakami J. The colored Jones polynomials and the simplicial volume of a knot. Acta Math. 186 (2001) 85-104
46. Murakami H., and Yokota Y. The colored Jones polynomials of the figure-eight knot and its Dehn surgery spaces (2004). math.GT/0401084
47. Murakami J., and Yano M. On the volume of a hyperbolic and spherical tetrahedron. Comm. Anal. Geom. 13 (2005) 379-400
48. Neumann W.D., and Yang J. Invariants from triangulations of hyperbolic 3-manifolds. Electron. Res. Announc. Amer. Math. Soc. 1 (1995) 72-79
49. Neumann W.D., and Yang J. Bloch invariants of hyperbolic 3-manifolds. Duke Math. J. 96 (1999) 29-59
50. Neumann W.D., and Zagier D. Volumes of hyperbolic three-manifolds. Topology 24 (1985) 307-332
51. Ponzano G., and Regge T. Semiclassical limit of Racah coefficients. In: Bloch F. (Ed). Spectroscopic and Group Theoretical Methods in Physics (1968), North-Holland 1-58
52. Roberts J. Classical 6 j-symbols and the tetrahedron. Geom. Topol. 3 (1999) 21-66
53. Ruijsenaars S.N.M. First order analytic difference equations and integrable quantum systems. J. Math. Phys. 38 (1997) 1069-1146
54. R. Scharein, Knotplot. http://www.pims.math.ca/knotplot/
55. Shintani T. On a Kronecker limit formula for real quadratic fields. J. Fac. Sci. Univ. Tokyo 24 (1977) 167-199
56. Takahashi M. On the concrete construction of hyperbolic structures of 3-manifolds. Tsukuba J. Math. 9 (1985) 41-83
57. W.P. Thurston, The geometry and topology of three-manifolds, in: Lecture Notes in Princeton University, Princeton, 1980
58. Thurston W.P. Hyperbolic structures on 3-manifolds. I. deformation of acylindrical manifolds. Ann. Math. 124 (1986) 203-246
59. Turaev V.G., and Viro O.Y. State sum invariants of 3-manifolds and quantum 6 j symbols. Topology 31 (1992) 865-902
60. J. Weeks, SnapPea. http://www.geometrygames.org/SnapPea/
61. Witten E. Quantum field theory and the Jones polynomial. Comm. Math. Phys. 121 (1989) 351-399
62. Woronowicz S.L. Quantum exponential function. Rev. Math. Phys. 12 (2000) 873-920
63. Yokota Y. On the potential functions for the hyperbolic structures of a knot complement. In: Ohtsuki T., Kohno T., Le T., Murakami J., Roberts J., and Turaev V. (Eds). Invariants of Knots and 3-Manifolds (Kyoto 2001). Geometry and Topology Monographs vol. 4 (2002), Univ. Warwick 303-311
64. Yoshida T. The η-invariant of hyperbolic 3-manifolds. Invent. Math. 81 (1985) 473-514