# Verification report — AMR-105-0051 (Fenn–Ilyutko–Kauffman–Manturov, Problem 51: a free link with several minimal representatives)

Verification date: 2026-10-14.

**Verdict.** The note gives a **complete positive answer** to the problem as stated, already for free knots.
- **Settled.** The free knot 𝒦 of K1 = `a b a e c b d f d g e g c f` has exactly two minimal representatives up to
  isomorphism, K1 and K2 = `a b a c e b d f d e g c g f` (seven vertices each, related by one third move on the
  triangle (e, c, g)). The two-component free link ℒ of D1 = {`e 1 e 3 5 2 6 4`, `1 2 3 4 5 6`} has exactly two,
  D1 and D2 = {`e 5 2 6 e 4 1 3`, `1 2 4 3 5 6`} (Theorems A and B).
- **Proved in the text.** Parts (i)–(iii): the third move, the non-isomorphism (arcs 6/6 against 4/8 at the unique
  even chord; 1/5 against 3/3), and the lower bound of seven vertices via the parity bracket, whose invariance is
  proved from local tables (Theorem 3.5) together with a minimality criterion (Proposition 4.1).
- **Computer-assisted.** Part (iv), "exactly two": the statements (E1), (E2) of Section 6, established by two
  independent programs (verification runs A and B), repeated by the programs of the note, and established again by
  the verification run of the written note (run 2, with its own programs).
- **Not new as a method.** Kauffman and Manturov (arXiv:1207.0719v3, Corollary 3.4 and the remark right after it)
  gave a minimality criterion and remarked that diagrams related by a third move (of the unoriented type) have the
  same unoriented state; in arXiv:1406.7331 (Section 5) they observed that a diagram with a triangle is equivalent
  to its image under the third move and remarked that minimal diagrams "having some triangles" can be constructed.
  Their criterion, applied to the explicit diagram X1 = `0 1 2 0 3 2 4 1 5 3 6 5 4 6` and its image X2 (whose
  unoriented states are isomorphic), also answers the problem. The note's contribution is, to
  our knowledge, the first explicit written example with a self-contained proof of non-isomorphism, minimality
  and the exact number of minimal representatives.
- **Open.** Free links whose minimal representatives are not related by third moves (not treated).

The note is unrefereed.

## Statement checked
- **Primary source.** R. Fenn, D. P. Ilyutko, L. H. Kauffman, V. O. Manturov, "Unsolved problems in virtual knot
  theory and combinatorial knot theory", Banach Center Publ. 103 (2014) 9–61, doi:10.4064/bc103-0-1 (Crossref);
  arXiv:1409.2823 (v1, the only version). Read in the TeX source of v1 (archive fetched 2026-10-14, sha256
  `a628d90d7376282e3633469add7093beb9baa5d08d6a4cc70cbabe13b556b857`).
  - Section 8 "A List of Problems", subsection 8.1 "Problems in and related with virtual knot theory": the first
    enumerate has 60 top-level items; item 51 (TeX lines 2076–2079) asks to prove or disprove the conjecture on the
    non-uniqueness of the minimal representative of a free link, "there exists a free link having several minimal
    representatives". No figure, reference or further context.
  - Section 2.2: free knots and links are those corresponding to Gauss diagrams without signs and arrows, with
    references to Manturov's arXiv:0901.2214 and 0902.0127.
  - The archive contains an unused figure `non_uniq.eps` (dated 6 June 2014, not referenced in the TeX source).
- **Definitions.** Manturov, "On free knots and links", arXiv:0902.0127v1 (only version; read in full): Section 2
  (framed 4-graphs, unicursal components, the three moves, the increasing second move without conditions, free
  links and knots), Section 2.1 (smoothings), Section 3.1 (the space with graphs with free loops set to zero;
  Theorem 1, "trivial": unique irreducible representative), Section 5 (the parity conditions), Corollary 1 in
  Section 5.1 (minimal "with respect to the number of vertices"), Section 6 (the bracket {·} for two-component
  links, odd = crossing of two components; Theorem 4, label `mainthm2`, invariance, with a sketched proof).
- **Corpus record.** ulamai/UnsolvedMath, AMR-105-0051 (dataset version 1.6.0; upstream status `open`); the
  statement is the source sentence verbatim.

## Readings
| Reading | Answer | Where |
|---|---|---|
| Free link, minimal = fewest vertices, several = two non-isomorphic framed 4-graphs (the source's reading) | yes: 𝒦 and ℒ, exactly two each | Theorems A, B |
| The same for free knots (stronger) | yes: 𝒦 | Theorem A |
| Strong minimality ("lives inside" every representative, arXiv:1412.8691) | unique when it exists, so non-uniqueness cannot refer to it; 𝒦 and ℒ have no such representative | Section 2.3 (b) |
| Minimal representatives not related by third moves | not treated; in both examples the minimal representatives form one class under third moves | Remark 7.1 |
| Labelled diagrams, or planar virtual diagrams up to planar isotopy | trivial, excluded by the source (Gauss diagrams without arrows and signs; framed graphs in Manturov's papers; uniqueness up to isomorphism in Kauffman–Manturov's conjecture) | Section 2.3 (e) |
| Virtual or flat virtual links, classical knots | different subject, not considered | Section 2.3 (d) |

## Results in the paper
- **Lemmas 2.1–2.3.** Framed 4-graphs as families of Gauss words; the moves (deleting a loop letter; deleting a
  bigon pair; exchanging the letters on the three edges of a triangle) and the smoothings ({A, B} and A·rev(B) for
  W = vAvB; AB and A·rev(B) for vA, vB) in words, with derivations.
- **Lemma 3.2.** The Gaussian parity (knots) and the mixed parity (two-component links) satisfy the axioms
  (P0)–(P3) of Definition 3.1.
- **Lemma 3.3.** Unique R2-normal form (local confluence and induction), basis of the space Ṽ.
- **Theorem 3.5.** Invariance of the parity bracket {D} = Σ_s [D_s] for every parity; proof by the local tables of
  Table 1 (first move; second move with even and with odd vertices, parallel and antiparallel; third move with three
  even vertices and with exactly one), including second moves on one edge or a free loop and triangles whose
  external ends are joined directly.
- **Proposition 4.1.** If D has exactly one even vertex, one smoothing D¹ is R2-irreducible without free loops and the
  other has a nonzero class different from [D¹], every diagram of the link has more vertices than D¹, and every
  minimal representative has a smoothing isomorphic to D¹.
- **Proposition 4.2.** The candidate lists, with three types of un-smoothing (type (3), a vertex between two
  components, was added; only type (1) occurs in the applications). Used only as a cross-check.
- **Section 5.** {K1} = [T6] + [T4], T6 = `c b d f d g a b a f c g`, T4 = {`b d f d`, `f a b a`};
  {D1} = [G6] + [H2], G6 = {`1 4 6 2 5 3`, `1 2 3 4 5 6`}, H2 = {`1`, `6`, `1 6`}; the triangles; the arcs.
- **Section 6.** (E1): among the 5,283 chord diagrams with seven chords exactly K1, K2 have the bracket {K1}; (E2):
  among the 20,160 diagrams {X, 123456} that can carry the bracket {D1}, exactly the classes of D1, D2 have it.
- **Remarks 7.1–7.5.** Third-move classes; girth two (double edges a–b, d–f, e–g) and the triangle of K1,
  consistent with the girth ≥ 4 conjecture; the criterion of Kauffman and Manturov and the example X1, X2; a second
  free knot with exactly two minimal representatives (`a b a c d c e b f g d g e f`, `a b a c d e f b f c g d g e`);
  the unused figure of the source archive.

## Computations (sanity checks; programs and outputs in reproducibility/)
Parts (i)–(iii) do not depend on a computation; part (iv) does, through (E1) and (E2).
- **Programs of the note** (standard library only). `check_note.py` (2 s): the word formulas against a half-edge
  model on 12,063 random smoothings, every finite statement of Section 5, Table 1 and the controls (sums differ for
  a triangle with two even or three odd vertices), the lists of Proposition 4.2 (312 → 14 classes, 168 → 8
  classes, two pass in each), Remarks 7.3 and 7.5: 12,124 assertions hold. `exhaustive.py` (15 s): every move
  between chord diagrams with ≤ 7 chords (20,869 moves) and between two-component diagrams with ≤ 6 vertices (7,876
  moves), no change of the bracket; negative controls (false third move: 1,483 of 7,734; false parity: 1,037 of
  2,348); (E1), (E2); Remark 7.4. `make_figure.py`: Figure 1 generated from the Gauss words and read back from the
  coordinates in main.tex.
- **Programs with which the results were first obtained** (`original/`, about 2 min): local tables; 16,055 random
  moves on two-component diagrams (4,408 third moves), no change; a false move changes the bracket in 597 of
  1,726 cases; normal forms unique on 37,387 graphs; the candidate lists (also with NetworkX); no diagram with ≤ 6
  vertices has the bracket of K1 (11,465 labelled chord diagrams) or of D1 (123,670 two-component diagrams).
- **Run A** (about 40 s) and **run B** (about 40 min, mostly a breadth-first search and its control and the
  unoriented states up to eight chords): see the next section.
- **Run 2** (`independent_run_2/`, about 20 s, own library `fl2.py`): every finite statement of Section 5 and of
  Remarks 7.1–7.5, Table 1 from tangles with its controls, Lemmas 2.2, 2.3, 3.3 on random graphs, 3,644 random
  moves without change of the bracket, Burnside numbers (5,283; 19,045), (E1), (E2), Proposition 4.2, the
  isomorphism of the unoriented states of X1 and X2, and the chord diagrams of the unused figure read again from its
  path data: 90 checks, 0 failures.
- **Re-runs.** All programs of `original/`, `verification_run_A/` and `verification_run_B/` were run again from
  copies on 2026-10-14 (at most three at a time): all 44 recorded outputs were reproduced, 33 byte for byte and 11 up
  to the fields that report running times. The programs of the note were run again from the extracted
  archive with `run_note.sh`. Run 2 ran the package again from the extracted archive of the previous version of
  this package (all outputs reproduced up to running times) and, for the final package, the programs of the note
  and its own programs from the extracted final archive. Details in `reproducibility/RERUN_LOG.txt`.

## Independent verification runs
The results were first obtained, with proofs, in a written version with the programs of `original/`. Two
independent verification runs, both AI-assisted, followed (2026-10-14); they did not see each other, and each wrote
its own programs from the definitions without the code of the first version.

| Item | Run A | Run B |
|---|---|---|
| Statement, definitions, readings against the sources | CONFIRMED (FIKM and Manturov TeX; item 51 of 60 recounted) | CONFIRMED (same) |
| Word model, moves, smoothings | PROVED (re-derived); smoothings in a port model agree with the formulas on 2,820 cases | PROVED; 11,984 smoothings compared |
| Lemma 3.3 (normal forms) | PROVED; 1,454 graphs, all orders | PROVED; 5,000 graphs, all orders |
| Lemma 3.2 (parities) | PROVED | PROVED |
| Theorem 3.5 (invariance) | PROVED; tables recomputed; 3,204 random moves | PROVED; tables recomputed; all 28,745 moves in the stated ranges, 0 violations |
| Proposition 4.1 | PROVED | PROVED |
| Proposition 4.2 | PROVED for the applications | incomplete as a general statement (type (3) missing); fixed |
| Theorem A (i)–(iii), Theorem B (i)–(iii) | CONFIRMED (re-derived and recomputed) | CONFIRMED (recomputed) |
| Theorem A (iv), Theorem B (iv) | CONFIRMED by complete enumeration: 5,283 seven-chord classes; 20,160 arrangements | CONFIRMED by the bracket of all diagrams with ≤ 7 vertices (19,045 two-component classes with 7 vertices) |
| Known cases | published minimal diagrams it examined (Manturov's nine-chord knot, permutation diagrams of Boothby–Henrich–Leaf) have no triangle | — |
| Literature and novelty | CONFIRMED_WITH_FIXES: cite the "having some triangles" remark of Kauffman–Manturov | NEEDS FIX: their criterion with the explicit X1, X2 also answers the problem |
| Bibliography | CONFIRMED | one title (JKTR version of arXiv:1406.7331) |
| The first version's programs | — | all 15 recorded outputs reproduced byte for byte from a copy |

No run found an error in a proof of the note's statements.

**Corrections required by the runs**, all applied in the note (details in `audit/fixes_applied.md` of the working
folder):
1. (A1, B4) The remark of Kauffman and Manturov after their Theorem 3 (minimal diagrams "having some triangles"),
   their observation that a diagram with a triangle is equivalent to its third-move image, and the example X1, X2
   to which their criterion (Corollary 3.4 of arXiv:1207.0719v3, numbered "Corollary 4" in the first version of
   the note; corrected after run 2) applies, are stated; the contribution is described as the first explicit written
   example with a self-contained proof (abstract, Section 1.2, Remark 7.3, "Scope and priority").
2. (A2) The girth of K1 and K2 is two (double edges), not three (Remark 7.2).
3. (A3) Degenerate readings and the strong minimality of arXiv:1412.8691 are discussed (Section 2.3 (b), (e)).
4. (A4) The unused figure is described precisely (Remark 7.5).
5. (A5, B1) The split "(i)–(iii) proved, (iv) computer-assisted" is kept everywhere, and the question for minimal
   representatives not related by third moves is stated as open (abstract, Section 1, Remark 7.1).
6. (B2) Proposition 4.2 has the missing type (3); part (iv) is based on the complete enumeration (Section 6).
7. (B3) A computational invariance test of the Gaussian-parity knot bracket: all 20,869 moves between chord diagrams
   with ≤ 7 chords, 0 violations (`exhaustive.py`, Table 2).
8. (B5) The published title of the JKTR paper ("sl(3), C2 and G2") and the pages 9–61 of the problem list are used.
9. (B6) The command lines of the first version's programs, with their arguments, are given in the README.

## Verification run 2 (of the written note)
After the note was written, a further independent verification run, also AI-assisted, examined its text
(2026-10-14). It wrote its own library and programs before opening those of the note
(`reproducibility/independent_run_2/`, with a README), read the sources again in their arXiv source archives (and the PDF of
arXiv:1207.0719v3 for its numbering), re-derived every proof line by line, re-ran the package from an extracted copy of
the archive, and repeated the literature search.

| Area | Verdict of run 2 |
|---|---|
| Statement fidelity | CONFIRMED: item 51 of the 60 top-level items of Section 8.1 of arXiv:1409.2823v1 (recounted), identical to the corpus statement; Manturov's definitions (arXiv:0902.0127v1, Sections 2–7; theorem and corollary counters checked); the question is answered as posed, and its scope (minimal representatives not related by third moves left open) is stated correctly |
| Proofs | CONFIRMED: Lemmas 2.1–2.3, 3.2, 3.3, Theorem 3.5 with Table 1, Propositions 4.1, 4.2, Sections 5 and 6 re-derived; no error; in {1, 526, 1256} both {2,5} and {5,6} are bigons (the text says "where {2,5} is a bigon", which is correct) |
| Cited results | CONFIRMED except one number: the criterion of arXiv:1207.0719v3 is Corollary 3.4 (that version numbers by section), not Corollary 4; and its remark after Corollary 3.4 (R3-related diagrams of the unoriented type have the same unoriented state) was not credited. All quotations verbatim |
| Computations | CONFIRMED with its own programs (90 checks, 0 failures): all of Section 5, Remarks 7.1–7.5, Table 1, Lemmas 2.2, 2.3, 3.3 on random graphs, 3,644 random moves without change of the bracket, Burnside numbers 5,283 and 19,045, (E1), (E2) (132 classes among the 20,160 arrangements; 28 classes with seven odd vertices; 18,885 as printed), Remark 7.4, the isomorphism of the unoriented states of X1 and X2, and the chord diagrams of the unused figure read again from its path data |
| Re-run of the package | from the extracted archive: `run_note.sh` reproduced all four outputs byte for byte; `original/` 15 outputs identical; run A 10 outputs (2 up to running times); run B all outputs reproduced up to running times (details in RERUN_LOG.txt) |
| Novelty and credit | no prior explicit example found; the anticipation by Kauffman and Manturov is closer than the first text said (the remark after Corollary 3.4); the novelty claim ("first explicit written example with a self-contained proof", "the method is not new", no priority claimed) is no stronger than the evidence |
| Presentation | house style, abstract, Verification and Scope paragraphs and AI paragraph in order; all pages rendered and inspected |

**Corrections required by run 2, all applied:**
1. "[KM14, Corollary 4]" → "[KM14, Corollary 3.4]" (Sections 1.2, 7.3 twice, Scope and priority; bibliography note).
2. The remark of Kauffman and Manturov after their Corollary 3.4 is cited (abstract, Section 1.2, Remark 7.3, Scope
   and priority); Remark 7.3 records that the unoriented states of X1 and X2 are isomorphic, those of K1, K2 not.
3. "The uniqueness of the minimal diagram is known for special classes" → "proved, or asserted"; the invariant of
   Manturov (Dokl. Math. 2013) extended by Ilyutko and Manturov (JKTR 2020, abstract read) is mentioned.
4. Free knots as Turaev's Gauss words up to homotopy (Turaev, Proc. LMS 2007; Gibson, Math. Ann. 2011) are
   mentioned in the introduction, and this literature is included in the search account.
5. This run is recorded in the Verification paragraph, in Section 6, in Section 8 and Table 2, here, and in the
   reproducibility package (`independent_run_2/`, README, RERUN_LOG).
6. "What was read" and the search paragraph are updated.

## Relation to the literature, novelty and scope
- **Searches (14 October 2026).** In the first version and the two runs: arXiv API queries (free knots and links,
  minimal diagrams and representatives, parity brackets, framed 4-graphs, Gauss words and phrases, nanowords), the
  works citing the problem list (20 in OpenAlex) and citing Kauffman–Manturov, Boothby–Henrich–Leaf and Manturov's
  parity papers, zbMATH (titles with "free knot"), Crossref, and a few web searches; at the writing: the arXiv
  sources of the cited papers, Crossref for every DOI, zbMATH for one chapter; in run 2: arXiv API (free knots and
  links; minimal diagrams and representatives; Gauss words and phrases, nanowords; "almost classification"; Manturov
  and minimality; versions), the sources of Gibson 0902.0062v2, Turaev math/0503683v2 and Manturov 1508.05555v2,
  1501.05208, OpenAlex (works citing the problem list 20, Kauffman–Manturov 2014 13, Kauffman–Manturov 2015 7,
  Boothby–Henrich–Leaf 1; titles screened), zbMATH (84 records on free knots, Gauss words and parity in MSC 57,
  screened), Crossref (12 look-ups) and one web search. No example of a free link with two non-isomorphic minimal
  representatives and no statement that the problem is settled was found.
- **Closest sources.** Kauffman–Manturov, arXiv:1207.0719v3 (Quantum Topol. 5 (2014) 523–539), Corollary 3.4 and the
  remark after it (R3-related diagrams of the unoriented type have the same unoriented state, so the state does not
  determine the diagram); arXiv:1406.7331v2 (J. Knot Theory Ramifications 24 (2015) 1550031, "Graphical constructions for
  the sl(3), C2 and G2 invariants for virtual knots, virtual braids and free knots"), Section 5: the third-move
  remark, Conjecture 1 (girth ≥ 4), "a very little is known" on uniqueness, and the remark after Theorem 3. They
  give no explicit minimal diagram with a triangle and do not compare a diagram with its third-move image.
- **What was read.** The arXiv versions listed in the note ("Scope and priority"); the journal versions were not
  compared. Not accessible, not read: Manturov–Ilyutko, *Virtual Knots: The State of the Art* (2012);
  Ilyutko–Manturov–Nikonov, J. Math. Sci. 193 (2013); Manturov, Sb. Math. 201 (2010) and Dokl. Math. 88 (2013);
  Petit, *Virtual strings and free knots* (Encyclopedia of Knot Theory, 2021); of Ilyutko–Manturov, J. Knot Theory
  Ramifications 29 (2020) 2050063 (an invariant of free graph-links that recovers the representative under certain
  conditions) only the abstract was read. A remark on the problem could be contained in them. Turaev's
  "Topology of words" (arXiv math/0503683v2) and Gibson's "Homotopy invariants of Gauss words" (arXiv:0902.0062v2),
  where free knots appear as Gauss words up to homotopy, were searched by run 2 and contain no statement on the
  uniqueness of minimal representatives.
- **Caveats.** The invariance for the mixed parity is Manturov's Theorem 4 (re-proved in the note); the
  Gaussian-parity bracket with all smoothings is the specialisation a = 1 mod 2 of his even Kauffman bracket
  (arXiv:0912.5348, Theorem 6, stated for virtual knots). The note re-proves both. This negative search is not a proof
  of novelty, and no priority is claimed.
- **Scope.** Answers the problem in the sense of non-isomorphic minimal representatives; minimal representatives not
  related by third moves remain open.

## Public release and license
This paper, its source files, and this verification report are licensed under the Creative Commons Attribution
4.0 International License (CC BY 4.0): https://creativecommons.org/licenses/by/4.0/