Ropelength-Filtered Knot Densities

Abstract

We study scale-invariant $p$-densities of knot types in Euclidean three-space, defined as the infimum over representatives of the length divided by an $L^p$-type mean of the pairwise chord lengths; the exponent $p$ is any value greater than minus one, including infinity, where the mean is the diameter. The unconstrained density is independent of the knot type. For exponents at most two, its common value is the density of the round circle, by the sharp mean-chord inequality; at the diameter endpoint it equals two; and for finite exponents greater than two it reduces to a planar convex extremal problem, in which the round circle fails to be extremal beyond an explicit threshold. To prevent this degeneration, which is caused by knotting at arbitrarily small scale, we introduce a ropelength filtration: at each level $\lambda$, the infimum is restricted to representatives whose ropelength is at most $\lambda$ times the ropelength of the knot type. The filtered density has minimizers, detects the unknot for exponents at most two, and converges to the unconstrained density as $\lambda$ tends to infinity. It is continuous in the exponent, including the diameter endpoint, and right-continuous in the filtration parameter. As a numerical test, we evaluate the density on published approximately tight configurations of the granny knot and the square knot. The sampled values differ by about 2.5 percent at exponent two, whereas their ordering is reversed at the diameter endpoint. These computations provide evidence that the ropelength filtration retains geometric information lost by the unconstrained theory.

arXiv comment31 pages, 6 figures, 1 table. v4: title changed (formerly "Unconstrained and Ropelength-Windowed p-Densities of Knot Types"); exposition reorganized; numerical comparison of the granny and square knots added. Based on the author's talk at PPICTA 2026