  A theorem of Jorgensen and Thurston implies that the volume of a hyperbolic
3-manifold is bounded below by a linear function of its Heegaard genus.
Heegaard surfaces and bridge surfaces often exhibit similar topological
behavior; thus it is natural to extend this comparison to ask whether a
$(g,b)$-bridge surface for a knot $K$ in $S^3$ carries any geometric
information related to the knot exterior. In this paper, we show that (unlike
in the case of Heegaard splittings) hyperbolic volume and genus $g$ bridge
numbers are completely independent. That is, for any $g$, we construct explicit
sequences of knots with bounded volume and unbounded genus $g$ bridge number,
and explicit sequences of knots with bounded genus $g$ bridge number and
unbounded volume.
