On complexity of three-dimensional hyperbolic manifolds with geodesic boundary

The nonintersecting classes ? _p,q are defined, with p, q ?? ? and p ?? q ?? 1, of orientable hyperbolic 3-manifolds with geodesic boundary. If M ?? ? _p,q , then the complexity c(M) and the Euler characteristic ??(M) of M are related by the formula c(M) = p???(M). The classes ? _q,q , q ?? 1, and ?_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 ?_3,1 and obtained exact values of complexity for these manifolds. The method of proof is based on calculating the ?-invariants of manifolds.