Algebraic & Geometric Topology

Polynomial $6j$–symbols and states sums

Nathan Geer and Bertrand Patureau-Mirand

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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 [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 h1H1(M,), homology class h2H2(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,ϕH1(M,)) 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,)H1(M,).

Article information

Algebr. Geom. Topol., Volume 11, Number 3 (2011), 1821-1860.

Received: 13 July 2010
Accepted: 8 October 2010
First available in Project Euclid: 19 December 2017

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Zentralblatt MATH identifier

Primary: 57M27: Invariants of knots and 3-manifolds 81Q99: None of the above, but in this section

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


Geer, Nathan; Patureau-Mirand, Bertrand. Polynomial $6j$–symbols and states sums. Algebr. Geom. Topol. 11 (2011), no. 3, 1821--1860. doi:10.2140/agt.2011.11.1821.

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