Polynomial $6j$–symbols and states sums

Abstract

For a given $2r$–th root of unity $\xi$, we give explicit formulas of a family of $3$–variable Laurent polynomials $J_{i,j,k}$ with coefficients in $\mathbb{Z}\left[\xi \right]$ that encode the $6j$–symbols associated with nilpotent representations of $U_{\xi }\left(\mathfrak{s}\mathfrak{l}\left(2\right)\right)$. For a given abelian group $G$, we use them to produce a state sum invariant ${\tau }^r\left(M,L,h_1,h_2\right)$ of a quadruplet (compact $3$–manifold $M$, link $L$ inside $M$, homology class $h_1\in H_1\left(M,\mathbb{Z}\right)$, homology class $h_2\in H_2\left(M,G\right)$) 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 $\tau \left(M,L,\varphi \in H^1\left(M,{ℂ}^{\ast }\right)\right)$ defined by the authors and Turaev in [arXiv:0910.1624] from the category of nilpotent representations of $U_{\xi }\left(\mathfrak{s}\mathfrak{l}\left(2\right)\right)$. They refine them as $\tau \left(M,L,\varphi \right)={\sum }_{h_1}{\tau }^r\left(M,L,h_1,\tilde{\varphi }\right)$ where $\tilde{\varphi }$ correspond to $\varphi$ with the isomorphism $H_2\left(M,{ℂ}^{\ast }\right)\simeq H^1\left(M,{ℂ}^{\ast }\right)$.