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

The tightest knot is not necessarily the smallest