  In recent years, several families of hyperbolic knots have been shown to have
both volume and $\lambda_1$ (first eigenvalue of the Laplacian) bounded in
terms of the twist number of a diagram, while other families of knots have
volume bounded by a generalized twist number. We show that for general knots,
neither the twist number nor the generalized twist number of a diagram can
provide two-sided bounds on either the volume or $\lambda_1$. We do so by
studying the geometry of a family of hyperbolic knots that we call double coil
knots, and finding two-sided bounds in terms of the knot diagrams on both the
volume and on $\lambda_1$. We also extend a result of Lackenby to show that a
collection of double coil knot complements forms an expanding family iff their
volume is bounded.
