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

We study a family of scale-invariant $p$-densities of knot types in $R^3$, defined as the ratio of length to an $L^p$-type spread of pairwise distances along a curve. The first point of the paper is that the unconstrained theory has a strong degeneration. Local knotting shows that, for every $p\in(-1,\infty]$ and every knot type $K$, the unconstrained $p$-density of $K$ is no larger than that of the unknot. Using the sharp mean-chord inequality of Exner--Harrell--Loss, we show that this degeneration is complete throughout the range $-1<p\le2$: for $p\ne0$ one has $$\rho_p(K)= \pi\left( \frac{\pi}{\int_0^\pi \sin^p\theta\,d\theta} \right)^{1/p},$$ while $\rho_0(K)=2\pi$. At the endpoint $p=\infty$, one also has $\rho_\infty(K)=2$ for every knot type $K$. The remaining finite range $p>2$ is analytically different: the round circle is not the relevant extremal curve in general, and knot-type independence in this range is left as a separate extremal problem. These degenerations motivate a constrained refinement. We introduce ropelength-windowed $p$-densities by imposing the thickness normalization $Thi(\gamma)\ge 1$ and the length bound $len(\gamma)\le \lambda Rop(K)$. These constraints prevent the collapse caused by arbitrarily small local knotting. We prove basic monotonicity properties and an existence theorem for minimizers of the ropelength-windowed problem. We also retain the polygonal approximation theorem for the unconstrained densities, showing that the continuous and polygonal theories agree asymptotically as the number of edges tends to infinity. The paper concludes with a list of open questions concerning the finite high-exponent range, constrained density spectra, thickness-controlled polygonal approximation, and regularized inverse-power extensions.