Ropelength-Filtered Knot Spaces

Abstract

Ropelength is usually studied as a minimisation problem for one knot at a time. Here we instead use ropelength to filter the space of all realisations of a knot type. For a knot type $K$ and budget $\Lambda$, let $Y_{\Lambda }(K)$ be the space of unit-thickness $C^{1,1}$ configurations of length at most $\Lambda$, modulo rigid motions and constant-speed reparametrisation. We prove compact capture for smooth knot families, yielding homotopical and homological exhaustion of the ordinary knot space by finite ropelength levels. In the normalised Euclidean model, every $Y_{\Lambda }(K)$ is compact; connected components are right-continuous and their merge levels are attained, giving an ultrametric on the components of the ideal stratum. We also prove that $Y_L(K)$ strongly deformation retracts onto its exact-length shell, so path connectivity at a fixed level is fixed-length physical isotopy. The connected merge scale and the path merge scale, the min--max quantity for constrained deformation paths, take the same values, although only the former is known to be attained. Moreover, all configurations in any fixed finite sublevel become path connected to one another at some common finite higher level, even if the original sublevel has infinitely many components. For higher homotopy we introduce ropelength widths and relate them to known models of spaces of knots. We record mirror symmetry and prove finite factorization across every positive filtration gap in a marked weak model, yielding all-degree homological q-tameness. Finally, we relate the path filtration to the companion swept-area construction: the infimal budget for admissible endpoint connections agrees with the path merge threshold. The resulting framework separates geometric, topological, and physical-isotopy questions and provides a quantitative foundation for the study of ropelength-filtered knot spaces.

arXiv comment50 pages, 18 figures, 4 tables