Geometric densities and compression radii of knot types

We study scale-invariant geometric quantities associated with embedded closed curves in Euclidean three-space, with an emphasis on their behavior under optimization within a fixed knot type. Given a Euclidean-invariant and scale-covariant size functional $D$, we define the $D$-density of a curve $\gamma$ by $\len(\gamma)/D(\gamma)$, the $D$-compression radius by $D(\gamma)/\Thi(\gamma)$, and the corresponding packing ratio as its reciprocal. For a single representative, ropelength factors as the product of the $D$-density and the $D$-compression radius. The main point is not this formal cancellation, but the separation it suggests after optimization: the density, compression, packing, and ropelength problems generally have different minimizing sequences. We develop this factorization framework for general scale-covariant size functionals. We prove the basic optimized inequality, give a criterion for equality after optimization, and compute the unknot case for the diameter and the minimal enclosing radius. We also prove polygonal approximation results for compression radii when $D=\diam$ and when $D=R_{\min}$, using standard convergence properties of polygonal thickness, and formulate the corresponding hypotheses for other $L^p$-type size functionals. Finally, we discuss relations with distortion, trunk, and supertrunk. The framework is intended as a structural companion to density-type invariants, rather than as an immediate source of stronger ropelength lower bounds. In particular, the optimized factorization by itself does not yield new ropelength bounds; such bounds require independent estimates for the density and compression factors.