Year: 2026
Type: preprint
Abstract: We prove relative smoothing with uniform reach control for one-parameter families of positive-reach $C^{1,1}$ knots and combine it with relative multijet transversality to obtain projection-generic regularization inside strict ropelength sublevels. Passing to a lifted Reidemeister multigraph reconstructs the admissible $π_0$ of every strict projection-framed ropelength sublevel, and the right-relaxed diagrammatic merge scale equals the geometric one. The normalised projection-framed sublevels are compact, so connected merge levels are attained. These degree-zero results are organized by a fibre cosheaf: a filtered relational cellular cosheaf of sets on the planar Reidemeister multigraph, whose stalks are strict fibre components and whose costalks are the relations realized by Reidemeister movies. The lifted multigraph is its Grothendieck construction, reconstruction is a descent statement, and the diagram image is its support. The diagram image computes merges exactly when no stalk is persistently separated, and a canonical flat coarsening is the coarsest diagram-indexed model that computes components. For amphichiral knots the spherical diagram image cannot detect mirror barriers, whereas the planar image can; no planar type of the standard diagrams of $4_1$ and $6_3$ is fixed by crossing switch. Symmetric hubs and finite certificates give upper bounds for mirror merges, and sign separators give lower bounds. For finite recognition we distinguish full rooted-ball occurrence, visibility in the monotone diagram-image filtration, and saturation. Every finite Reidemeister submultigraph becomes visible at a finite ropelength level, a visible saturated Barbensi-Celoria characteristic certificate determines the knot type up to mirroring, and a universal computable crossing-complexity bound controls visibility.
Source: arXiv (Cornell University)
Author: Ozawa, Makoto
Institution: Komazawa University
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7160373700