Unconstrained and Ropelength-Windowed $p$-densities of Knot Types

Abstract

We study scale-invariant $p$-densities of knot types in $\mathbb R^3$, defined as length divided by an $L^p$-type mean of pairwise chord lengths. The unconstrained density is independent of the knot type for every $p\in(-1,\infty]$. For $-1 2$, the common value reduces to a planar convex extremal problem. Comparison with the doubly covered segment gives the explicit upper bound $p_*=3.5720244135\ldots$ for the circle-breaking exponent and rules out circle optimality for $p>p_*$. We then restrict to representatives satisfying $\operatorname{Rop}(\gamma)\le\lambda\operatorname{Rop}(K)$. The resulting windowed density has minimizers and detects the unknot throughout the sharp mean-chord range: $\rho^{\operatorname{rop}}_{p,\lambda}(K)=c_p$ if and only if $K=U$, for $-1<p\le2$ and $\lambda\ge1$. It converges to the unconstrained density as $\lambda\to\infty$. We also prove continuity in $p$ on $(-1,\infty]$, right-continuity in $\lambda$, an $O((\lambda\operatorname{Rop}(K))^{2/3})$ upper bound at $p=\infty$, polygonal approximation, and universal strict fixed-edge polygonal gaps.