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 $t{r}_{\epsilon }$ and the real-valued Turaev - Viro type invariant $t{v}_{\epsilon }$. These two invariants for 3-manifolds are related by the equality $|t{r}_{\epsilon }{|}^2\cdot (\epsilon +2)=t{v}_{\epsilon }$, where $\epsilon$ is a root of the equation ${\epsilon }^2=\epsilon +1$. Finally, we show that $t{v}_{\epsilon }$ coincides with the well-known $\epsilon$ invariant for 3-manifolds.

28 pages, 38 figures, 22 bibliography items. In the current version, the introduction and bibliography have been expanded