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 $\mathcal{G}<sup>{\mathrm{lift}}_{Λ,u}(K)$, for $Λ\ge \mathrm{Rop}(K)$, recording diagram data and Reidemeister moves that lift to admissible thick deformations below the ropelength level $Λ$. 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 $C<sup>{1,1}$ ropelength-sublevel space are conditional on explicitly isolated projection--Cerf tameness and coherent finite-pattern thick-movie liftability hypotheses.

• The paper defines ropelength-filtered lifted Reidemeister graphs and finite recognition length, measuring when a finite diagrammatic pattern identifies a knot up to mirroring.
• The methodology combines Barbensi–Celoria’s finite-local graph invariants with projection–Cerf theory, while separating unconditional graph results from polygonal-model and analytic conditional results.
• The framework relates diagrammatic merge scales to geometric deformation persistence under tameness assumptions, with trefoil recognition and coherent thick movie liftability remaining open quantitative problems.

This paper, (2605.03350) by Makoto Ozawa, develops a diagrammatic counterpart to the author's earlier ideal-stratum and deformation-persistence framework for knot types. The central object is a family of ropelength-filtered lifted Reidemeister graphs $G_{\Lambda,u}^{\operatorname{lift}}(K)$, indexed by a ropelength bound $\Lambda$ and a projection direction $u$, which records only those diagrams and Reidemeister transitions realizable by thickness-one representatives of length at most $\Lambda$. The paper's main contribution is the definition of the finite recognition length $L_{\operatorname{char},u}(K)$ — the first scale at which a finite Reidemeister pattern characterizing $K$ up to mirroring becomes visible in this filtered graph — together with an explicit and careful separation of which parts of the theory are unconditional, which are proved in controlled models, and which remain conditional on stated analytic hypotheses.

Geometric setting: ideal strata and ropelength sublevel spaces

The paper works with $C^{1,1}$ embedded curves in $\mathbb{R}^3$ whose thickness is Federer's reach, equivalently the minimum of the curvature radius and half the doubly-critical self-distance. Ropelength is $Rop(\gamma)=Len(\gamma)/Thi(\gamma)$; after fixing thickness one, the sublevel space $X_\Lambda(K)$ consists of representatives of $\Lambda$0 of length at most $\Lambda$1, modulo orientation-preserving isometries. Its path components are admissible components, and the ideal stratum $\Lambda$2 collects ropelength-minimizing representatives.

A notable convention choice is made here: the paper uses the exact slice $\Lambda$3 rather than the sublevel condition $\Lambda$4. The author explicitly concedes that the two conventions may have different path-component structures, since rescaling paths inside $\Lambda$5 can connect regions that stay separated in the slice. The slice convention is preferred because it fixes scale and prevents arbitrarily small local knotting from appearing at negligible geometric size — a point that matters directly for the recognition-scale philosophy.

Graph-theoretic input and a no-go principle

The unconditional backbone of the paper is Barbensi–Celoria's theorem that the $\Lambda$6-Reidemeister graph $\Lambda$7 is a complete invariant up to mirroring, together with their Corollary 5.4 giving a finite-local form: for every vertex diagram $\Lambda$8, some finite rooted ball $\Lambda$9 already characterizes the knot type. Ozawa packages this as rooted typed balls $u$0, where the edge typing (Reidemeister type, local orientation, tentacle classes) is intrinsically recoverable from local graph structure via their Theorem 3.23, so no extra decoration is required.

From this the paper derives a no-go principle (Proposition in Section 3): any embedding $u$1 preserving rooted local Reidemeister structure on a characterizing neighborhood forces $u$2 or $u$3. Consequently, a preorder on knot types defined by inclusion of completed Reidemeister graphs collapses to equality up to mirroring. This observation justifies the paper's methodological shift: rather than comparing full graphs, one studies the growth process by which finite characteristic patterns first become visible under the ropelength filtration. This is precisely what $u$4 measures.

Projection–Cerf mechanism and its analytic status

Diagram changes along admissible paths are governed by crossings of the projection discriminant $u$5, whose codimension-one pieces are the $u$6 (cusp-type), $u$7 (self-tangency), and $u$8 (triple-point) walls, in line with classical perestroika theory of plane curves. The paper defines vertex birth scales $u$9, edge birth scales $\Lambda$0, and a ropelength–projection Cerf graphic recording when walls become reachable at each length level.

The honest analytical core of the paper lies in what is and is not proved:

• Unconditional: the finite-local graph theory of Barbensi–Celoria.
• Proved in controlled models: a finite-dimensional bookkeeping lemma (which, notably, assumes the Whitney-stratified discriminant structure rather than proving it) and a fully unconditional polygonal projection–Cerf theorem. The latter uses Rawdon's polygonal thickness and Tarski–Seidenberg quantifier elimination to show the thick polygon space is semialgebraic, so the discriminant admits a finite semialgebraic Whitney stratification whose codimension-one strata are exactly the PL analogues of the three Reidemeister events.
• Conditional: component reconstruction $\Lambda$1 and the agreement of geometric and diagrammatic merge scales hold only under the projection–Cerf tameness assumption, which the author states plainly is not proved for the full infinite-dimensional $\Lambda$2 space, where thickness is a nonsmooth constraint. The paper identifies concrete obstacles: perturbations may cross the boundary $\Lambda$3 or $\Lambda$4, and discriminant strata can interact with reach constraints when resolving a tangency simultaneously decreases a doubly-critical distance.

Finite recognition length

The central definitions are pattern visibility and recognition length. A finite typed pattern $\Lambda$5 is visible at level $\Lambda$6 if it admits a coherent lift to $\Lambda$7 — shared vertices of $\Lambda$8 must map to the same lifted vertices, so visibility is genuinely a statement about graph patterns, not about separately realizable vertices and edges. Then

$\Lambda$9

over all finite characteristic patterns $L_{\operatorname{char},u}(K)$0, is the first scale at which enough local Reidemeister structure exists to identify $L_{\operatorname{char},u}(K)$1 up to mirroring.

The main result, finite recognizability, is a conditional theorem: assuming coherent finite-pattern thick Reidemeister liftability (Assumption on thick movies with common endpoint representatives, fixed boundary collars, and clearance from exterior strands), every knot type satisfies $L_{\operatorname{char},u}(K)$2. The proof combines the two independent inputs: Barbensi–Celoria supplies the characteristic pattern unconditionally; the liftability hypothesis supplies its realization at finite level. The author is explicit that this hypothesis "packages the main unresolved geometric coherence problem" and cannot be obtained from independent local movie insertions, because cycles and shared endpoints in the pattern require endpoint coherence. The theorem is also qualitative: it gives no effective bound on $L_{\operatorname{char},u}(K)$3, and estimating or minimizing $L_{\operatorname{char},u}(K)$4 is identified as the quantitative problem behind the whole framework.

Relation to persistence and examples

Under tameness, diagrammatic merge scales agree with geometric merge scales from the ideal-stratum theory, so the diagrammatic merge tree is an exact $L_{\operatorname{char},u}(K)$5-level shadow of deformation persistence — but a coarse one, forgetting birth scales, radii, diameters, and characteristic patterns. Growth invariants introduced include the Reidemeister radius $L_{\operatorname{char},u}(K)$6 from the ideal vertex set, the diagrammatic diameter (which need not be monotone in $L_{\operatorname{char},u}(K)$7, since new edges create shortcuts), and window-dependent crossing profiles.

Two concrete cases anchor the theory. For the unknot, Fenchel's theorem combined with the curvature bound gives $L_{\operatorname{char},u}(K)$8 with unique minimizer (the unit circle), so the ideal merge tree is trivial; nevertheless the growth graph is nontrivial, and the first liftable $L_{\operatorname{char},u}(K)$9 event requires positive ropelength beyond $K$0 — its exact value is posed as an open problem. For the trefoil, known bounds give $K$1; a separated local-curl model suggests the heuristic first-$K$2 scale $K$3, but the author carefully notes this model covers only a subclass of liftable events and yields neither a lower nor an upper bound for unrestricted first visibility, since non-separated kinks may interact with nearby strands and global reach.

Limitations and open problems

The paper is candid about its conditional structure, and the following problems define its frontier:

• Full projection–Cerf tameness for the $K$4 ropelength space remains unproved; a quantitative reach-preserving perturbation theorem compatible with both the discriminant and the length bound would be needed.
• Coherent thick movie insertion — the finite-pattern liftability hypothesis itself — is the key unresolved geometric input.
• Quantitative recognition: computing $K$5, including whether the first visible trefoil $K$6 event occurs below $K$7.
• The finite witness principle (Conjecture): for standard classes of invariants and decompositions, finite witness scales $K$8 exist and satisfy $K$9. The paper stresses this conjecture is programmatic, restricted to "standard" structures informally understood, and not a claim about arbitrary set-theoretic invariants.

Conclusion

The paper establishes a coherent finite-diagrammatic layer over the ideal-stratum program: filtered lifted Reidemeister graphs translate geometric persistence into wall-crossing graph growth, and finite recognition length measures when a knot's identity becomes finitely visible at bounded ropelength. Its rigor is stratified honestly — unconditional graph theory, unconditional polygonal models, and conditional $C^{1,1}$0 statements under two explicitly isolated hypotheses. Whether those hypotheses hold, and whether recognition scales can be computed even for the trefoil, are the questions on which the viability of the broader finite-witness principle depends.