On complexity of three-dimensional hyperbolic manifolds with geodesic boundary

The nonintersecting classes $\mathscr H_{p,q}$ are defined, with $p,q\in\mathbb N$ and $p\ge q\ge1$, of orientable hyperbolic $3$-manifolds with geodesic boundary. If $M\in\mathscr H_{p,q}$, then the complexity $c(M)$ and the Euler characteristic $\chi(M)$ of $M$ are related by the formula $c(M)=p-\chi(M)$. The classes $\mathscr H_{q,q}$, $q\ge1$, and $\mathscr H_{2,1}$ are known to contain infinite series of manifolds for each of which the exact values of complexity were found. There is given an infinite series of manifolds from $\mathscr H_{3,1}$ and obtained exact values of complexity for these manifolds. The method of proof is based on calculating the $\varepsilon$-invariants of manifolds.