Invariants for links and 3-manifolds from the modular category with two simple objects

Abstract

We describe the simplest non-trivial modular category $\mathfrak{E}$ with two simple objects. Then we extract from this category the invariant for non-oriented links in 3-sphere and two invariants for 3-manifolds: the complex-valued Turaev - Reshetikhin type invariant $tr_{\varepsilon}$ and the real-valued Turaev - Viro type invariant $tv_{\varepsilon}$. These two invariants for 3-manifolds are related by the equality $|tr_{\varepsilon}|^2\cdot (\varepsilon + 2) = tv_{\varepsilon}$, where $\varepsilon$ is a root of the equation $\varepsilon^2 = \varepsilon + 1$. Finally, we show that $tv_{\varepsilon}$ coincides with the well-known $\varepsilon$ invariant for 3-manifolds.