Geometry & Topology

$6j$–symbols, hyperbolic structures and the volume conjecture

Francesco Costantino

Full-text: Open access


We compute the asymptotical growth rate of a large family of Uq(sl2) 6j–symbols and we interpret our results in geometric terms by relating them to volumes of hyperbolic truncated tetrahedra. We address a question which is strictly related with S Gukov’s generalized volume conjecture and deals with the case of hyperbolic links in connected sums of S2×S1. We answer this question for the infinite family of fundamental shadow links. Corrections  The paper was republished with corrections on 19 October 2007.

Article information

Geom. Topol., Volume 11, Number 3 (2007), 1831-1854.

Received: 15 January 2007
Revised: 24 August 2007
Accepted: 25 July 2007
First available in Project Euclid: 20 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M27: Invariants of knots and 3-manifolds
Secondary: 57M50: Geometric structures on low-dimensional manifolds

Jones polynomial volume conjecture hyperbolic volume $6j$–symbol quantum invariant


Costantino, Francesco. $6j$–symbols, hyperbolic structures and the volume conjecture. Geom. Topol. 11 (2007), no. 3, 1831--1854. doi:10.2140/gt.2007.11.1831.

Export citation


  • S Baseilhac, R Benedetti, Quantum hyperbolic invariants of $3$–manifolds with ${\rm PSL}(2,\mathbb C)$–characters, Topology 43 (2004) 1373–1423
  • S Baseilhac, R Benedetti, Classical and quantum dilogarithmic invariants of flat ${\rm PSL}(2,\mathbb{C})$–bundles over $3$–manifolds, Geom. Topol. 9 (2005) 493–569
  • S Baseilhac, R Benedetti, Quantum hyperbolic geometry (2006)
  • F Costantino, Colored Jones invariants of links in $\#S^2 \times S^1$ and the volume conjecture (2007) to appear in J. of the London Math. Soc.
  • F Costantino, D P Thurston, $3$–manifolds efficiently bound $4$–manifolds (2005)
  • P Doyle, G Leibon, 23040 symmetries of hyperbolic tetrahedra (2003)
  • R Frigerio, C Petronio, Construction and recognition of hyperbolic $3$–manifolds with geodesic boundary, Trans. Amer. Math. Soc. 356 (2004) 3243–3282
  • S Garoufalidis, T Le, Asymptotics of the colored Jones function of a knot (2007)
  • S Gukov, Three-dimensional quantum gravity, Chern–Simons theory, and the A–polynomial, Comm. Math. Phys. 255 (2005) 577–627
  • R M Kashaev, The hyperbolic volume of knots from the quantum dilogarithm, Lett. Math. Phys. 39 (1997) 269–275
  • R M Kashaev, O Tirkkonen, A proof of the volume conjecture on torus knots, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 269 (2000) 262–268, 370 Translation in J. Math. Sci. 115 (2003), 2033-2036
  • J Milnor, Hyperbolic geometry: the first 150 years, Bull. Amer. Math. Soc. (N.S.) 6 (1982) 9–24
  • Y Mohanty, The Regge symmetry is a scissors congruence in hyperbolic space, Algebr. Geom. Topol. 3 (2003) 1–31
  • H Murakami, J Murakami, The colored Jones polynomials and the simplicial volume of a knot, Acta Math. 186 (2001) 85–104
  • H Murakami, J Murakami, Asymptotic behaviors of the colored Jones polynomial of a torus knot (2004)
  • H Murakami, J Murakami, M Okamoto, T Takata, Y Yokota, Kashaev's conjecture and the Chern–Simons invariants of knots and links, Experiment. Math. 11 (2002) 427–435
  • H Murakami, Y Yokota, The colored Jones polynomial of the figure eight knot and its Dehn surgery spaces
  • J Murakami, M Yano, On the volume of a hyperbolic and spherical tetrahedron, Comm. Anal. Geom. 13 (2005) 379–400
  • W D Neumann, D Zagier, Volumes of hyperbolic three-manifolds, Topology 24 (1985) 307–332
  • J Roberts, Classical $6j$–symbols and the tetrahedron, Geom. Topol. 3 (1999) 21–66
  • Y U Taylor, C T Woodward, $6j$ symbols for $U\sb q(sl_2)$ and non-Euclidean tetrahedra, Selecta Math. (N.S.) 11 (2005) 539–571
  • W P Thurston, The geometry and topology of three manifolds
  • V G Turaev, Quantum invariants of knots and $3$–manifolds, de Gruyter Studies in Mathematics 18, Walter de Gruyter & Co., Berlin (1994)
  • A Ushijima, A volume formula for generalised hyperbolic tetrahedra, from: “Non-Euclidean geometries”, Math. Appl. (N. Y.) 581, Springer, New York (2006) 249–265
  • R Van der Veen, The volume conjecture for Whitehead chains (2006)
  • H Zheng, Proof of the volume conjecture for Whitehead doubles of a family of torus knots (2005)