For a given –th root of unity , we give explicit formulas of a family of –variable Laurent polynomials with coefficients in that encode the –symbols associated with nilpotent representations of . For a given abelian group , we use them to produce a state sum invariant of a quadruplet (compact –manifold , link inside , homology class , homology class ) with values in a ring related to . 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 –manifold , the invariants are related to defined by the authors and Turaev in [arXiv:0910.1624] from the category of nilpotent representations of . They refine them as where correspond to with the isomorphism .
"Polynomial $6j$–symbols and states sums." Algebr. Geom. Topol. 11 (3) 1821 - 1860, 2011. https://doi.org/10.2140/agt.2011.11.1821