  We consider hyperbolic links that admit alternating projections on surfaces
in compact, irreducible 3-manifolds. We show that, under some mild hypotheses,
the volume of the complement of such a link is bounded below in terms of a
Kauffman bracket function defined on link diagrams on the surface.
  In the case that the 3-manifold is a thickened surface, this Kauffman bracket
function leads to a Jones-type polynomial that is an isotopy invariant of
links. We show that coefficients of this polynomial provide 2-sided linear
bounds on the volume of hyperbolic alternating links in the thickened surface.
As a corollary of the proof of this result, we deduce that the twist number of
a reduced, twist reduced, checkerboard alternating link projection with disk
regions, is an invariant of the link.
