Geometric densities and compression radii of knot types

TL;DR

This paper introduces a framework for analyzing geometric quantities of knots, focusing on their behavior under optimization, and explores the relationships between density, compression, and ropelength.

Contribution

It develops a factorization framework for scale-covariant size functionals, providing inequalities, criteria for equality, and polygonal approximation results for knot analysis.

Findings

Proved an optimized inequality relating density and compression.

Computed the unknot case for diameter and minimal enclosing radius.

Established polygonal approximation results for compression radii.

Abstract

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.