  We introduce a generalization of the Krushkal polynomial to nonorientable
surfaces, and prove that this polynomial has a natural quasi-tree expansion.
This generalized Krushkal polynomial contains the Bollob\'as-Riordan polynomial
of a (possibly nonorientable) ribbon graph as a specialization. The quasi-tree
expansion proven here then extends the recent quasi-tree expansions of the
Bollob\'as-Riordan polynomial deduced in the oriented case by A. Champanerkar
et al. and in the more general unoriented case by E. Dewey and F.
Vignes-Tourneret. The generalized Krushkal polynomial also contains the Las
Vergnas polynomial of a cellulation of a surface as a specialization; we use
this fact to deduce a quasi-tree expansion for the Las Vergnas polynomial.
