Abstract
The invariant of a link in three-sphere, associated with the cyclic quantum dilogarithm, depends on a natural number N. By the analysis of particularexamples, it is argued that, for a hyperbolic knot (link), the absolute valueof this invariant grows exponentially at large N, the hyperbolic volume of the knot (link) complement being the growth rate.
This is a preview of subscription content, log in via an institution to check access.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
Alexander, J. W.: Topological invariants of knots and links, Trans. Amer. Math. Soc. 30(1928), 275–306.
Jones, V. F. R.: A polynomial invariant for knots and links via Von Neumann algebras, Bull. Amer. Math. Soc. 12(1985), 103–111.
Yang, C. N.: Some exact results for the many-body problem in one dimension with repulsive delta-function interaction, Phys. Rev. Lett. 19(1967), 1312–1314.
Baxter, R. J.: Partition function of the eight-vertex lattice model, Ann. Phys. 70(1972), 193–228
Thurston, W.: The geometry and topology of hyperbolic 3-manifolds, Lecture notes, Princeton University, Princeton 1977/78.
Adams, C., Hildebrand, M. and Weeks, J.: Hyperbolic invariants of knots and links, Trans. Amer. Math. Soc. 1(1991), 1–56.
Adams, C.: The Knot Book. An Elementary Introduction to the Mathematical Theory of Knots, W. H. Freeman, New York, 1994.
Witten, E.: Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121(1989), 351–399.
Atiyah, M.: Topological quantum field theories, Publ. Math. IHES 68(1989), 175–186.
Witten, E.: 2 + 1-dimensional gravity as an exactly soluble system, Nuclear Phys. B. 311(1988/89), 46–78.
Milnor, J. W.: Hyperbolic geometry: the first 150 year, Bull. Amer. Math. Soc. 6(1) (1982), 232–248.
Faddeev, L. D. and Kashaev, R. M.: Quantum dilogarithm, Modern Phys. Lett. A 9(1994), 427.
Faddeev, L.D.: Discrete Heisenberg–Weyl group and modular group, Lett.Math. Phys. 34(1995), 249–254.
Faddeev, L. D.: Current-like variables in massive and massless integrable models, Lectures delivered at the International School of Physics ‘Enrico Fermi’, held in Villa Monastero, Varenna, Italy, 1994, hep-th/9408041
Kashaev, R. M.:Quantum dilogarithmas a 6j-symbol,Modern Phys. Lett. A 9(1994), 3757–3768.
Kashaev, R. M.: A link invariant from quantum dilogarithm, Modern Phys. Lett. A 10(1995), 1409–1418.
Bazhanov, V. V. and Reshetikhin, N. Yu.: Remarks on the quantum dilogarithm, J. Phys. A,Math. Gen. 28(1995), 2217–2226.
Bazhanov, V. V. and Baxter, R. J.: Star-triangle relation for a three-dimensional model, J. Statist. Phys. 71(1993), 839.
Kashaev, R. M., Mangazeev, V. V. and Stroganov, Yu. G.: Star-square and tetrahedron equations in the Baxter-Bazhanov model, Internat. J. Modern Phys. A 8(1993), 1399.
Turaev, V. G. and Viro, O. Y.: State sum invariants of 3-manifolds and quantum 6j-symbols, Topology 31(1992), 865–902.
Kirillov, A. N.: Dilogarithm identities, 1994, hep-th/9408113.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
KASHAEV, R.M. The Hyperbolic Volume of Knots from the Quantum Dilogarithm. Letters in Mathematical Physics 39, 269–275 (1997). https://doi.org/10.1023/A:1007364912784
Issue Date:
DOI: https://doi.org/10.1023/A:1007364912784