  Dasbach and Lin proved a "volumish theorem" for alternating links. We prove
the analogue for alternating link diagrams on surfaces, which provides bounds
on the hyperbolic volume of a link in a thickened surface in terms of
coefficients of its reduced Jones-Krushkal polynomial. Along the way, we show
that certain coefficients of the 4-variable Krushkal polynomial express the
cycle rank of the reduced Tait graph on the surface.
