Finite Knot Theory via Ropelength-Filtered Reidemeister Graphs

Abstract

This paper develops a form of finite knot theory as a diagrammatic sequel to the ideal-stratum and deformation-persistence framework for knot types. Thick representatives in bounded ropelength sublevel spaces are studied through the finite Reidemeister data visible in generic projections. For each projection direction $u$, we introduce the ropelength-filtered lifted Reidemeister graphs $\mathcal{G}^{\mathrm{lift}}_{\Lambda,u}(K)$, for $\Lambda\ge \mathrm{Rop}(K)$, recording diagram data and Reidemeister moves that lift to admissible thick deformations below the ropelength level $\Lambda$. Using the finite-local reconstruction theorem of Barbensi--Celoria, we define characteristic Reidemeister patterns and the finite recognition length $L_{\mathrm{char},u}(K)$, the first ropelength scale at which a finite pattern recognizing $K$, up to mirroring, appears in the lifted graph. The finite-local graph-theoretic part is unconditional; finite-dimensional and polygonal models provide controlled settings; the corresponding statements for the full $C^{1,1}$ ropelength-sublevel space are conditional on explicitly isolated projection--Cerf tameness and coherent finite-pattern thick-movie liftability hypotheses.