Homologically trivial part of the Turaev-Viro invariant order 7

Abstract

A homologically trivial part of any Turaev-Viro invariant odd order $r$ is a Turaev-Viro type invariant order $\frac{r + 1}{2}$. In this paper we find an explicit formulas for this Turaev -- Viro type invariant, corresponding to the invariant order $r = 7$. Our formulas express $6j$-symbols and colour weights in terms of $\gamma$, where $\gamma$ is a root of the polynomial $\mathcal{T}(x) = x^3 - 2x^2 - x + 1$.