  We provide an upper bound on the Cheeger constant and first eigenvalue of the
Laplacian of a finite-volume hyperbolic 3-manifold M, in terms of data from any
surgery diagram for M. This has several consequences. We prove that a family of
hyperbolic alternating link complements is expanding if and only if they have
bounded volume. We also provide examples of hyperbolic 3-manifolds which
require 'complicated' surgery diagrams, thereby proving that a recent theorem
of Constantino and Thurston is sharp. Along the way, we find a new upper bound
on the bridge number of a knot that is not tangle composite, in terms of the
twist number of any diagram of the knot. The proofs rely on a theorem of Lipton
and Tarjan on planar graphs, and also the relationship between many different
notions of width for knots and 3-manifolds.
