  For $D$ a reduced alternating surface link diagram, we bound the twist number
of $D$ in terms of the coefficients of a polynomial invariant. To this end, we
introduce a generalization of the homological Kauffman bracket defined by
Krushkal. Combined with work of Futer, Kalfagianni, and Purcell, this yields a
bound for the hyperbolic volume of a class of alternating surface links in
terms of these coefficients.
