Abstract
In the present article, we describe a modular category \(\mathfrak {E}\) with exactly two simple objects. We use a specialtechnique and derive two invariants from \(\mathfrak {E}\) ;namely, a complex-valued invariant \(rt_{\varepsilon }\) ofReshetikhin–Turaev type (for unoriented links in a \(3\) -sphere and for \(3\) -manifolds) and a real-valued invariant \(tv_{\varepsilon}\) of Turaev–Viro type (for \(3\) -manifolds). The values of these two invariants for \(3\) -manifolds satisfy the equality \(|rt_{\varepsilon }|^2\cdot(\varepsilon + 2) = tv_{\varepsilon }\) , where \(\varepsilon\) is a root of the equation \(\varepsilon ^2 = \varepsilon +1\) . We prove that the invariant \(tv_{\varepsilon }\) coincides with the \(\varepsilon\) -invariant for \(3\) -manifolds.
