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.