The tightest knot is not necessarily the smallest

TL;DR

This paper investigates the relationship between minimal-length knot configurations and their convex hull volume, providing counterexamples where the ideal form does not minimize volume, highlighting complex geometric behaviors.

Contribution

It presents counterexamples to the conjecture that minimal-length knots also minimize convex hull volume, using T(p,2) torus knots and analyzing local minima during length annealing.

Findings

Certain knots have non-volume-minimizing ideal configurations.

The $8_{19}$ knot's global minimum volume configuration is non-ideal.

Local minima in convex hull volume can arise from buckling and symmetry breaking.

Abstract

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.