Invariants of Links and $$3$$-Manifolds Arising From a Modular Category withTwo Simple Objects

In the present article, we describe a modular category $$\mathfrak {E} $$ with exactly two simple objects. We use a special technique and derive two invariants from $$\mathfrak {E}$$ ; namely, a complex-valued invariant $$rt_{\varepsilon }$$ of Reshetikhin–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.