Geometric densities and compression radii of knot types

Abstract

We introduce scale-free compression radii and packing ratios of knot types and clarify their relation to geometric densities. Let $D$ be a Euclidean-invariant, scale-covariant size functional on embedded closed curves. For a curve $\gamma$, we define the $D$-density by $\operatorname{Len}(\gamma)/D(\gamma)$, the $D$-compression radius by $D(\gamma)/\operatorname{Thi}(\gamma)$, and the corresponding packing ratio as its reciprocal. For each representative, ropelength is 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 within a fixed knot type: density, compression, and ropelength generally have different minimizing sequences. We establish the basic optimized inequality and a criterion for equality, and compute the unknot case for diameter and minimal enclosing radius. We also prove polygonal approximation theorems for the compression radii associated with these two size functionals, using standard convergence properties of polygonal thickness, and formulate sufficient hypotheses for analogous results 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 alone does not yield new ropelength bounds; such bounds require independent estimates for the density and compression factors.