Abstract:
In this paper we construct a modular category $\mathfrak{E}$ containing exactly two simple objects. Using a special technique, two invariants are extracted from it: a complex-valued invariant of Reshetikhin – Turaev type $rt_{\varepsilon}$ of unoriented links in the $3$-sphere and of $3$-manifolds, and a real-valued invariant of Turaev – Viro type $tv_{\varepsilon}$ of $3$-manifolds. The values of these two invariants of $3$-manifolds are related by the equality $|rt_{\varepsilon}|^2\cdot (\varepsilon + 2) = tv_{\varepsilon}$, where $\varepsilon$ is the root of the equation $\varepsilon^2 = \varepsilon + 1$. It is proved that the $tv_{\varepsilon}$ invariant exactly coincides with the $\varepsilon$-invariant of $3$-manifolds.
Key words:modular category, Reshetikhin – Turaev type invariant, Turaev – Viro type invariant, $\varepsilon$-invariant.
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