  Weakly generalised alternating knots are knots with an alternating projection
onto a closed surface in a compact irreducible 3-manifold, and they share many
hyperbolic geometric properties with usual alternating knots. For example,
usual alternating knots have volume bounded above and below by the twist number
of the alternating diagram due to Lackenby. Howie and Purcell showed that a
similar lower bound holds for weakly generalised alternating knots. In this
paper, we show that a generalisation of the upper volume bound does not hold,
by producing a family of weakly generalised alternating knots in the 3-sphere
with fixed twist number but unbounded volumes. As a corollary, generalised
alternating knots can have arbitrarily small cusp density, in contrast with
usual alternating knots whose cusp densities are bounded away from zero due to
Lackenby and Purcell. On the other hand, we show that the twist number of a
weakly generalised alternating projection does gives two sided linear bounds on
volume inside a thickened surface; we state some related open questions.

Geometric Topology