2021/11/16 by Alexander Klotz, Klotz, Alexander R.
Biochemistry, Genetics and Molecular Biology · Mathematics · #Connective tissue disorders research #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2111.08218
openalex publication_date 2021/11/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 819 has a non-ideal global minimum in its convex hull volume even when the thickness of its tube is taken into account.