# Verification report — AMR-103-0230 (Stoimenow: determinants of prime alternating achiral knots, Problem 12.25 of the Ohtsuki list)

Verification date: 2026-10-19.

**Verdict.** The note gives a **partial answer**; the question remains open. Every statement the note makes is
proved, and the scope is as follows.
- **Proved.** (Theorem 1.1) For every plane network X with m ≥ 2 terminals, the graph G(X) = X ∪ ψ(X*) (X glued to
  its dual network turned over by a half-turn ψ of S²) is self-dual; if it is nonseparable with an odd number of
  spanning trees, its alternating diagram is a prime alternating achiral knot of determinant τ(G(X)). For m = 2,
  det = τ(X)² + F(X)²; for stars with m = 3, 4, det = |Λ3|², |Λ4|² (Proposition 3.2). (Theorem 1.2) Realized
  determinants are closed under n ↦ s⁴n (s ≥ 3 odd) and n ↦ 3⁶n; so every j⁴m (j odd, m > 1 composed of primes
  ≡ 1 mod 4) and every 3^(2e)m (e ≥ 2) is realized, and the question reduces to n = d²m with d > 1 a squarefree product of primes ≡ 3 mod 4.
  (Theorem 1.3) 5q² is realized for every prime q ≡ 3 mod 4. (Theorem 1.4) An explicit three-terminal range
  d²(A² + B²). (Theorem 1.5, computation) Every n ≤ 10^6 not covered by Stoimenow's theorems (16,703 numbers).
  (Proposition 7.1) No network with nonseparable closure has a pair (q, qj) with q prime.
- **Known, not claimed.** The problem, Theorems 4.1, 4.2, 4.4 and Proposition 4.1 of Stoimenow (2005), his
  formulas (9) and (12) (the cases m = 2, 3), and his remark that 5p², p ≤ 11, is realized on the polyhedron 8*.
  Gluing a graph to its dual to obtain self-dual maps is classical (Smith–Tutte 1950; Archdeacon–Richter 1992;
  Servatius–Christopher 1992; Servatius–Servatius 1996); Tutte (1979) studied spanning trees of self-dual maps.
- **Open.** All n = d²m > 10^6 (d > 1 squarefree, product of primes ≡ 3 mod 4; m > 1 composed of primes ≡ 1 mod 4)
  not covered by Theorems 1.3, 1.4 and their scalings. Section 7 explains why the three families used do not
  settle them.

The note is unrefereed.

## Statement checked
- **Primary source.** T. Ohtsuki (ed.), "Problems on invariants of knots and 3-manifolds", Geom. Topol. Monogr. 4
  (2002) 377–572, doi:10.2140/gtm.2002.4.377; Problem 12.25 (A. Stoimenow) with its Remark on p. 541 (PDF page 169
  of the publisher's file). The corpus statement matches the source. The Remark says that the achiral knot of
  determinant n can be chosen alternating or prime, "but not always both", and that any further exception beyond
  1, 9, 49 would be larger than 2000 and not a square.
- **Paper behind the problem.** A. Stoimenow, "Square numbers, spanning trees and invariants of achiral knots",
  Comm. Anal. Geom. 13 (2005) 591–631, doi:10.4310/CAG.2005.v13.n3.a5; read as arXiv:math/0003172v3 (26 February
  2003), Sections 1–6. Conjecture 4.2 is the problem; Conjecture 6.1 is its graph form. Definitions used:
  achiral = isotopic to the mirror image, without orientation conditions (Sec. 1); det = |Δ(−1)| (Sec. 2); for an
  alternating diagram det = number of spanning trees of the checkerboard graph (Lemma 6.1); Menasco's primeness
  criterion as quoted in Sec. 4.2; reduced alternating diagrams have minimal crossing number (Kauffman, Murasugi,
  Thistlethwaite, as quoted in Sec. 4.2).

## Readings
| Reading | Answer | Where |
|---|---|---|
| Achirality with any behaviour of the orientation (the source's notion) | used throughout | Section 1.1 |
| "Prime" excludes the unknot; "alternating" = has an alternating diagram | used; the knots are given by reduced alternating diagrams | Lemma 2.2 |
| n with no prime factor ≡ 3 mod 4, or n an odd square | settled by Stoimenow (Theorems 4.2, 4.4 of the 2005 paper) | Section 1.1 |
| The residual set R (odd, sum of two squares, not a square, q² ∣ n for a prime q ≡ 3 mod 4) | realized up to 10^6; open above, except the families of Theorems 1.2–1.4 | Theorems 1.2–1.5 |
| Graph form (Conjecture 6.1 of the 2005 paper) | its "if" part holds for n ≤ 10^6 | Remark 6.1 |
| Links (even n), unknotting number one, strong achirality | not treated | — |

## Results in the paper
- **Lemmas 2.1–2.8.** Checkerboard diagrams, primeness of D(G) for nonseparable G (with Menasco's theorem), the
  boundary-region lemma, dual networks and double duality, the substitution formula (Lemma 2.6), the series and
  parallel rules (Corollary 2.7), and p(N*) = (F(N), τ(N)) (Lemma 2.8).
- **Theorem 1.1 and Proposition 3.2.** Self-duality of G(X) under the half-turn ψ with ψ(t_j) = μ_{−j},
  ψ(μ_k) = t_{−k}; achirality by rotating the knot through the half-turns about the axis of ψ; Remark 3.1 for
  orientation-reversing self-dualities. Wheel formulas: Λ3 = x1·conj(z0)·z2 + y1·Im(z0z2) (Stoimenow's (12)),
  Λ4 = x1x3 z0z2 + x1y3y2 z0 + y1x3y0 z2 + y1y3(z0z2 + y0y2); W_2, W_3 = K_4 and W_4 have 5, 16 and 45 spanning
  trees.
- **Theorem 1.2, Lemma 4.1.** Gadgets Z_s (pair (s², s²)) and Z' (pair (27, 27)); substitution of an edge of X
  multiplies τ(G(X)) by k².
- **Theorem 1.3, Lemma 5.1.** 4-wheels with Λ4 = (1 + 2i)Q for Q = a² + ab + b² or Q = 3Y² − X²; every prime
  q ≡ 3 mod 4 has such a representation (Thue's lemma and a descent with the unit 2 ± √3).
- **Theorem 1.4.** 3-wheels with Λ3 = d(A + iB).
- **Theorem 1.5.** 16,703 explicit certificates (15,330 of type B ∥ B*, 771 three-terminal and 602 four-terminal
  wheels, all with series-parallel networks).
- **Proposition 7.1, Corollary 7.2.** q ∤ F(B) for τ(B) = q prime (critical group and its pairing; injectivity of
  the Abel–Jacobi map, Baker–Norine Theorem 1.8); hence q²(1 + c²) with 1 + c² uniquely represented is not of the
  form B ∥ B* (e.g. 45, 245, 2313). A self-duality fixing an edge forces τ even.

## Computations (programs and outputs in reproducibility/)
Only Theorem 1.5 and the finite identities of Proposition 3.2 depend on computation.
- **Program of the note** (`check_note.py`, standard library, about 20 s, TOTAL failures: 0): residual set;
  exact polynomial identities and substitutions; series/parallel rules, duals and gadgets on explicit graphs;
  Lemma 5.1 for all 39,322 primes q ≡ 3 mod 4 below 10^6; Theorem 1.4 on 3,000 parameter sets; all certificates by
  tree sums and by Λ; Theorem 1.1 on 400 random plane networks (m = 2..6, not only series-parallel) and 150 stars,
  with a control gluing; the plane maps of all 16,703 certificates (Euler characteristic, Kirchhoff τ = n,
  nonseparable, orientation-preserving isomorphism to the dual map, one component); Theorem 1.2 on 180
  substitutions; Proposition 7.1 on 300 random graphs.
- **Programs with which the results were first obtained** (`original/`): `verify_certs.py` (all 16,703 certificates,
  seconds), map checks of 3,371 certificates, sympy identities, the Friedl–Miller–Powell condition on 400
  certificates, the generator (about 4 GB of memory). `planemap.py` uses the clockwise dart order for duals, so the
  orientation labels of its isomorphism test are interchanged; its programs test both orientations, either of
  which suffices for achirality.

## Independent verification runs
Two independent verification runs, both AI-assisted, examined the first write-up of the results, each with its own
programs. Run A: statement, definitions, all proofs, the computations. Run B: independent computation (own map
library, all certificates), literature and novelty. Then the note's own program `check_note.py` was written and run
for the final text, and a third independent verification run (verification run 2 on the written note, also
AI-assisted) checked the final text; see the next section.

| Item | Run A | Run B | Program of the note |
|---|---|---|---|
| Statement and definitions against the sources | CONFIRMED | CONFIRMED | — |
| Lemmas 2.1–2.8 (Facts A–D) | CONFIRMED (re-derived) | CONFIRMED (re-derived) | rules, duals and gadgets tested |
| Theorem 1.1 (self-duality, achirality, both orientations) | CONFIRMED (re-derived; maps of 50 networks and 1,500 certificates, failing control) | CONFIRMED (530 non-series-parallel networks, all certificates) | CONFIRMED (400 networks, 150 stars, all certificates) |
| Proposition 3.2 (Λ3, Λ4; tree counts 16, 45) | CONFIRMED (sympy, own tree enumeration) | CONFIRMED (sympy, brute-force trees) | CONFIRMED (exact polynomials) |
| Theorem 1.2 | CONFIRMED (24 map instances) | CONFIRMED (300 map instances) | CONFIRMED (180 map instances) |
| Theorem 1.3 and Lemma 5.1 | CONFIRMED (primes below 20,000; maps for 32 primes) | CONFIRMED (all primes below 10^6; maps for q < 3000) | CONFIRMED (all primes below 10^6) |
| Theorem 1.4 | CONFIRMED (40 map instances) | CONFIRMED (400 map instances) | CONFIRMED (3,000 parameter sets) |
| Theorem 1.5 | CONFIRMED (own parser, tree sums; R(10^6) = certificate set) | CONFIRMED (all 16,703 maps in about 7 s; writhe 0; Friedl–Miller–Powell condition for 17,301 pairs (n, q)) | CONFIRMED (tree sums; all 16,703 maps) |
| Proposition 7.1 | CONFIRMED (re-derived) | consistent (random test) | consistent (300 graphs) |
| Programs of `original/` | read; outputs consistent | re-run from copies; outputs byte-identical, certificates regenerated identically | — |
| Status | PARTIAL, CONFIRMED | CONFIRMED_WITH_FIXES (PARTIAL) | — |

Neither run found an error in a statement or a proof. Corrections required by the runs and made in the note:
Stoimenow's remark on 5p², p ≤ 11, and the polyhedron 8* (the 4-wheels are the self-dual members of the 8*
knots; Theorem 1.3 extends the remark to all q); the literature on self-dual maps and Tutte (1979) instead of a
novelty claim for the construction; the words "but not always both" attributed to the Remark in the problem list;
the tree counts 16 (W_3) and 45 (W_4); full references for Menasco, Kauffman, Murasugi, Thistlethwaite and
Baker–Norine; the orientation convention of `planemap.py`; the follow-up paper in Math. Z. judged by its title only.

## Verification run 2 on the text of the note (third independent run)
A further independent verification run, AI-assisted, checked the text of the note line by line, as if nothing had
been checked before. Its programs and outputs are in `reproducibility/independent_run_2/`.
- **Statement fidelity: CONFIRMED.** It re-read the problem list (p. 541) and Stoimenow, arXiv:math/0003172v3
  (Sections 1, 2, 4.1–4.3 and 6, with formulas (9), (12), the remark after (12), Theorems 4.1–4.4, Proposition 4.1
  and Conjectures 4.2 and 6.1). The problem, the definitions and the list of known results agree with the sources,
  and the scope (a partial answer; the question remains open) is stated correctly.
- **Proofs: CONFIRMED.** It re-derived every proof:
  - Lemmas 2.1–2.8;
  - the self-duality ψ(G) = G* (ψ(t_j) = μ_{−j}, ψ(μ_k) = t_{−k}, distinct faces F_k by Lemma 2.3);
  - achirality by the rotations ψ_t, and primeness via Menasco;
  - Proposition 3.2, Lemma 4.1 and Theorem 1.2;
  - Lemma 5.1 (Thue's lemma, reciprocity, the descent with 2 ± √3) and Theorems 1.3 and 1.4;
  - Proposition 7.1 (critical group, pairing, injectivity), Corollary 7.2, and the remark on edges fixed by a
    self-duality.

  It found no mathematical error.
- **Computations: CONFIRMED.** Its own programs, written from the text of the note, include a plane-map library whose
  dual orientation it validated geometrically. With them it checked:
  - 500 random plane networks (m = 2..6, 367 not series-parallel) and all 16,703 certificate maps: Euler
    characteristic 2, an orientation-preserving isomorphism G(X) → G(X)*, Kirchhoff count, nonseparability and one
    component; the control gluing μ_k ↦ t_{k+1} is non-planar for every m ≥ 3;
  - the residual set (16,703; 39 below 2000), and every certificate by tree sums over W_3, W_4 independent of the Λ
    formulas;
  - exact symbolic identities, and 300 gadget substitutions;
  - Lemma 5.1 and Theorem 1.3 for all 39,322 primes q ≡ 3 mod 4 below 10^6, Theorem 1.4 on 3,000 parameter sets
    and in 200 maps, and Proposition 7.1 on 300 random graphs;
  - the count of 973,177 series-parallel pairs.

  It re-ran every program of the package from an extracted copy of the source archive. All outputs were reproduced
  byte for byte apart from timing lines, and the generator regenerated `certificates.csv` identically.
- **Novelty: no printed answer found.** It searched the arXiv API, OpenAlex (the 15 works citing Stoimenow 2005 and
  the 11 citing Stoimenow 2007), the zbMATH API, Crossref (all 16 DOIs verified) and made one web search. It also
  read the publisher's summary of Tutte (1979).
- **Corrections made in the note.** No statement or proof changed. The corrections were:
  - Stoimenow's remark that no alternating arborescent tangle has a pair (0,p), (p,0), (p,p) is now credited in
    Section 7;
  - n = 1 is attributed to Stoimenow §4.2, and 9, 49 to Proposition 4.1;
  - an argument was added that a nonseparable X ∥ X* has nonseparable X̂, so that Corollary 7.2 applies to
    Theorem 1.1(a);
  - the description of the search in Section 6 is corrected: no certificate pair entry exceeds 1000, and the
    "windows" were search ranges for the arms;
  - the reduction in the abstract now reads "d > 1 a squarefree product of primes ≡ 3 mod 4";
  - the wording of the proof of Theorem 1.2, of Remark 3.3 (publisher's summary of Tutte 1979, "central reflex")
    and of the reference to Conjecture 6.1 ("slightly stronger", in Stoimenow's words) was adjusted;
  - the Verification and Scope paragraphs now record this run.
- **Status: CONFIRMED (partial answer), with the fixes above applied.**

## Relation to the literature, novelty and scope
- **Searches (19 October 2026).** arXiv API (achiral, amphicheiral, amphichiral knots with determinants or
  spanning trees; self-dual graphs and spanning trees; the author of the problem), OpenAlex (15 works citing the
  2005 paper; 11 citing the 2007 paper) and Semantic Scholar (17), zbMATH, Crossref, and six web searches (five in
  the first two runs, one in verification run 2). No printed answer to the problem
  was found in these sources. Determinants of amphichiral knots appear as obstructions (e.g. Boyle–Issa 2024), and
  tables of strongly positive amphicheiral knots and classification results for small braid index exist; none
  realizes the numbers of R.
- **What was read.** The problem page of the Ohtsuki list; Stoimenow (2005), arXiv v3, Sections 1–6;
  Friedl–Miller–Powell, arXiv:1706.07940v3, Theorem 1.1; Baker–Norine, arXiv:math/0608360v3, Theorem 1.8. Known
  only from titles and bibliographic summaries: Smith–Tutte 1950, Archdeacon–Richter 1992, Servatius–Christopher
  1992, Servatius–Servatius 1996, Tutte 1979 (its publisher's summary, read in verification run 2); Stoimenow's
  Math. Z. paper (2007) by its title (and, in run B, its
  zbMATH review, on leading coefficients of polynomial invariants). Menasco, Kauffman, Murasugi, Thistlethwaite and
  Hartley–Kawauchi are used as quoted by Stoimenow. All DOIs were checked with Crossref.
- **Caveats.** Run B could not read Archdeacon–Richter and Servatius–Christopher; whether the hemisphere gluing
  appears there in this form was not checked, and the note presents it as a general form of a known construction
  principle. The identification of the three-terminal wheels with Stoimenow's diagrams (12) was checked at the level
  of the determinant formulas, not of the diagrams. This negative search is not a proof of priority.
- **Scope.** The note answers the problem positively for n ≤ 10^6 and for the infinite families of Theorems
  1.2–1.4; the general case remains open. No priority is claimed for the construction principle.

## 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/