Swept-Area Metrics on Ropelength-Filtered Knot Spaces

Abstract

This paper develops an intrinsic area geometry on ropelength-filtered spaces of embedded curves. For a knot type $K$ and a ropelength level $\Lambda$, admissible isotopies pass through representatives of thickness at least one and length at most $\Lambda$. Their swept area is the parametrized area traced by the moving curve. Minimizing this quantity, with endpoint alignment optimized over orientation-preserving reparametrizations and Euclidean isometries, defines an extended distance on the corresponding moduli space. The first main result is that this distance is a genuine extended metric, not merely a pseudometric. The trace of every admissible isotopy defines an integral $2$-current whose boundary is the difference of the endpoint $1$-currents. The Euclidean-orbit flat distance between these currents is therefore a lower bound for swept-area distance and separates distinct moduli classes. On each admissible component the metric is finite. Closed admissible isotopies also define a swept-area length function on a regularity-restricted admissible fundamental group, with quantitative control under change of basepoint. We derive computable calibration bounds from the projected-area vector, including a Euclidean-invariant lower bound on the quotient space. These give exact distance formulas for concentric round unknots and for homothetic ellipses under explicit admissibility hypotheses. We also prove rigidity of the ideal unknot and a labelled finite-dimensional metric estimate on uniformly non-collinear polygonal strata. Finally, we relate the metric to filtered topology, merge scales, density and compression radius, and swept-area weighted lifted Reidemeister graphs. The rigorous diagrammatic comparison is formulated for diagrammatically generic isotopies, with comparison to the unrestricted metric obtained under an explicit generic-approximation hypothesis.

Major revision. The swept-area construction is now proved to define a genuine extended metric using a flat-norm lower bound for integral currents. The title has been changed from Pseudometrics to Metrics