Swept-Area Pseudometrics on Ropelength-Filtered Knot Spaces

TL;DR

This paper introduces swept-area pseudometrics on knot spaces filtered by ropelength, providing new tools for measuring isotopy complexity and establishing non-degeneracy and rigidity results.

Contribution

It defines a novel swept-area pseudometric framework on knot spaces, including fundamental group structures and exact distance formulas for specific knot types.

Findings

Non-degeneracy proven on finite-dimensional polygonal strata.

Exact distance formulas obtained for concentric unknots and ellipses.

Rigidity established for the ideal unknot.

Abstract

We introduce swept-area pseudometrics on ropelength-filtered spaces of knot representatives. For a knot type \(K\) and a ropelength level \(\Lambda\), admissible isotopies are required to pass through curves of thickness at least one and length at most \(\Lambda\). The swept area is the parametrized area traced by the moving curve, and its infimum over admissible isotopies defines an extended pseudometric on each admissible component. We also define the admissible fundamental group of a based admissible component and equip it with a swept-area length function. The construction is separated from the rigidity questions it raises. The zero-distance quotient is always a metric space, while non-degeneracy before quotienting is treated separately. We prove non-degeneracy on uniformly non-collinear finite-dimensional polygonal strata. We also prove calibration lower bounds from projected signed area, including a rotation-invariant supremum over oriented planes, and use them to obtain exact distance formulas for concentric round unknots and homothetic planar ellipses. We further prove rigidity of the ideal unknot. The framework is related to static scale-free invariants such as density and compression radius, and to filtered-topological structures such as ideal strata and merge scales. We define swept-area weighted lifted Reidemeister graphs and prove that, for diagrammatically generic isotopies, the associated diagrammatic distance is bounded above by the geometric swept-area distance. We also record monotonicity in the ropelength parameter and formulate problems toward full non-degeneracy and approximation theory.