We show that the cusp volume of a hyperbolic alternating knot can be
bounded above and below in terms of the twist number of an alternating
diagram of the knot. This answers a question asked by Thistlethwaite
on the cusp geometry of these knots. In addition to giving
diagrammatical estimates on cusp volume, this also leads to geometric
estimates on lengths of slopes, in terms of a diagram of the knot.
All these estimates are explicit. This is joint work with Marc
Lackenby.