Homologically trivial part of the Turaev-Viro invariant order 7

TL;DR

This paper derives explicit formulas for a specific homologically trivial part of the Turaev-Viro invariant at order 7, expressing key components in terms of roots of a cubic polynomial.

Contribution

It provides the first explicit formulas for the homologically trivial part of the Turaev-Viro invariant at order 7, linking $6j$-symbols and color weights to a cubic polynomial root.

Findings

Explicit formulas for $6j$-symbols and color weights at order 7

Connection between the invariant part and roots of a cubic polynomial

Advancement in understanding Turaev-Viro invariants at specific orders

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 $\mathcal{T}(x) = x^3 - 2x^2 - x + 1$.