  The notion of Gem-Matveev complexity has been introduced within
crystallization theory, as a combinatorial method to estimate Matveev's
complexity of closed 3-manifolds; it yielded upper bounds for interesting
classes of such manifolds. In this paper we extend the definition to the case
of non-empty boundary and prove that for each compact irreducible and
boundary-irreducible 3-manifold it coincides with the modified Heegaard
complexity introduced by Cattabriga, Mulazzani and Vesnin. Moreover, via
Gem-Matveev complexity, we obtain an estimation of Matveev's complexity for all
Seifert 3-manifolds with base $\mathbb D^2$ and two exceptional fibers and,
therefore, for all torus knot complements.
