Finite Knot Theory via Ropelength-Filtered Reidemeister Graphs

Abstract

We develop a finite-recognition framework for knot types using ropelength-filtered Reidemeister graphs. For a fixed projection direction, bounded-ropelength thick-knot spaces determine a lifted Reidemeister multigraph and a monotone diagram-image filtration in the classical Reidemeister multigraph. Finite decorated neighborhoods in this filtration serve as recognition certificates. We prove that every finite Reidemeister submultigraph becomes visible at a finite ropelength level, with a computable universal bound in terms of crossing complexity. Consequently, every knot type has finite recognition length, up to mirroring. We also characterize when a finite pattern admits a vertex-coherent geometric lifting by a finite face-consistency condition; this includes tree-shaped patterns and separated cube systems. For strict ropelength sublevels, a relative parametric smoothing theorem with uniform reach control, together with multijet transversality, yields component reconstruction by Reidemeister movies. Passing to arbitrarily small right relaxations recovers the geometric merge scales exactly. Thus the principal finite-recognition and merge-scale results do not require a global projection--Cerf tameness hypothesis.