Ropelength-Filtered Reidemeister Graphs

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. For families over a compact polyhedron we prove relative smoothing of the slices after an arbitrarily small budget increase. Passing to a lifted Reidemeister multigraph reconstructs the admissible ${\pi }_0$ of every strict projection-framed ropelength sublevel. We also compare the strong, quotient, and uniform-slack path relations and show that the corresponding path and admissible merge thresholds agree. The normalised projection-framed sublevels are direction-marked spaces over the normalised ropelength sublevels of arXiv:2604.17905; in particular they are compact Hausdorff, so connected merge levels are attained. At closed critical levels we use only arbitrarily small right relaxation and do not assert exact genericity within the critical level itself. For finite recognition we distinguish three notions that play different roles: full rooted-ball occurrence in the completed $S^2$-Reidemeister multigraph, visibility in the monotone diagram-image filtration, and saturation of a visible occurrence. Every finite Reidemeister submultigraph becomes visible at a finite ropelength level, and a visible saturated BC-characteristic certificate determines the knot type up to mirroring. A universal computable crossing-complexity bound controls visibility. Finally, vertex-coherent lifting of a finite pattern is characterized by a face-consistent planar labeling, in particular for tree-shaped patterns and separated cube systems.

arXiv comment64 pages, 12 figures