  In this note, we attempt to find counterexamples to the conjecture that the
ideal form of a knot, that which minimizes its contour length while respecting
a no-overlap constraint, also minimizes the volume of the knot, as determined
by its convex hull. We measure the convex hull volume of knots during the
length annealing process, identifying local minima in the hull volume that
arise due to buckling and symmetry breaking. We use T(p,2) torus knots as an
illustrative example of a family of knots whose locally minimal-length
embeddings are not necessarily ordered by volume. We identify several knots
whose central curve has a convex hull volume that is not minimized in the ideal
configuration, and find that $8_{19}$ has a non-ideal global minimum in its
convex hull volume even when the thickness of its tube is taken into account.
