Polynomial 6j–symbols and states sums

Geer, Nathan¹ ; Patureau-Mirand, Bertrand²

¹Mathematics & Statistics, Utah State University, 3900 Old Main Hill, Logan, UT 84322, USA
²LMAM, Universite de Bretagne-Sud, BP 573, F-56017 Vannes, France

Algebraic and Geometric Topology, Tome 11 (2011) no. 3, pp. 1821-1860

Résumé

For a given 2r–th root of unity ξ, we give explicit formulas of a family of 3–variable Laurent polynomials Ji,j,k with coefficients in ℤ[ξ] that encode the 6j–symbols associated with nilpotent representations of Uξ(sl(2)). For a given abelian group G, we use them to produce a state sum invariant τr(M,L,h1,h2) of a quadruplet (compact 3–manifold M, link L inside M, homology class h1 ∈ H1(M, ℤ), homology class h2 ∈ H2(M,G)) with values in a ring R related to G. The formulas are established by a “skein” calculus as an application of the theory of modified dimensions introduced by the authors and Turaev in [Compos. Math. 145 (2009) 196–212]. For an oriented 3–manifold M, the invariants are related to τ(M,L,ϕ ∈ H1(M, ℂ∗)) defined by the authors and Turaev in [arXiv:0910.1624] from the category of nilpotent representations of Uξ(sl(2)). They refine them as τ(M,L,ϕ) = ∑ h1τr(M,L,h1,ϕ̃) where ϕ̃ correspond to ϕ with the isomorphism H2(M, ℂ∗) ≃ H1(M, ℂ∗).

Détail

DOI : 10.2140/agt.2011.11.1821

Keywords: $6j$–symbols, state sum, skein calculus, quantum groups, $3$–manifolds

Bibliographie

[1] S Baseilhac, R Benedetti, Quantum hyperbolic invariants of $3$–manifolds with $\mathrm{PSL}(2,\mathbb C)$–characters, Topology 43 (2004) 1373

[2] F Costantino, J Murakami, On $\mathrm{SL}(2,\C)$ quantum $6j$–symbols and its relation to the hyperbolic volume.

[3] N Geer, B Patureau-Mirand, V Turaev, Modified $6j$–symbols and $3$–manifold invariants

[4] N Geer, B Patureau-Mirand, V Turaev, Modified quantum dimensions and re-normalized link invariants, Compos. Math. 145 (2009) 196

[5] C Kassel, Quantum groups, Graduate Texts in Math. 155, Springer (1995)

[6] A N Kirillov, N Y Reshetikhin, Representations of the algebra ${U}_q(\mathrm{sl}(2)),\;q$–orthogonal polynomials and invariants of links, from: "Infinite-dimensional Lie algebras and groups (Luminy-Marseille, 1988)" (editor V G Kac), Adv. Ser. Math. Phys. 7, World Sci. Publ. (1989) 285

[7] G Masbaum, P Vogel, $3$–valent graphs and the Kauffman bracket, Pacific J. Math. 164 (1994) 361

[8] T Ohtsuki, Quantum invariants. A study of knots, $3$–manifolds, and their sets, Series on Knots and Everything 29, World Sci. Pub. (2002)

[9] V G Turaev, Quantum invariants of knots and $3$–manifolds, de Gruyter Studies in Math. 18, de Gruyter (1994)

[10] V G Turaev, O Y Viro, State sum invariants of $3$–manifolds and quantum $6j$–symbols, Topology 31 (1992) 865

[11] O Y Viro, Quantum relatives of the Alexander polynomial, Algebra i Analiz 18 (2006) 63