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2011 Polynomial $6j$–symbols and states sums
Nathan Geer, Bertrand Patureau-Mirand
Algebr. Geom. Topol. 11(3): 1821-1860 (2011). DOI: 10.2140/agt.2011.11.1821


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,).


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Nathan Geer. Bertrand Patureau-Mirand. "Polynomial $6j$–symbols and states sums." Algebr. Geom. Topol. 11 (3) 1821 - 1860, 2011.


Received: 13 July 2010; Accepted: 8 October 2010; Published: 2011
First available in Project Euclid: 19 December 2017

zbMATH: 1232.57015
MathSciNet: MR2821443
Digital Object Identifier: 10.2140/agt.2011.11.1821

Primary: 57M27 , 81Q99

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

Rights: Copyright © 2011 Mathematical Sciences Publishers


Vol.11 • No. 3 • 2011
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