# Verification report — AMR-103-0226 (Ohtsuki's problem list, Problem 12.21; Shinohara's question)

Verification dates: 2026-10-02 (first independent verification run) and 2026-10-02/03 (second independent
verification run). Both runs were AI-assisted.

**Verdict.** The answer is yes. For every n = 4k + 1 with k > 0 there is a knot K in S^3 with det K = n and
σ(K) = 4 (and its mirror image has σ = −4). The knot can be taken of genus 2 unless n is a square all of whose
prime factors are ≡ 1 mod 4, and of genus at most 3 in that case. Hence {det K : σ(K) = 4} = 5 + 4N, answering
Stoimenow's Question 5.1 (Acta Arith. 129 (2007)), and a pair (d, s), d odd positive, s even, is (det K, σ(K)) for
some knot iff d ≡ (−1)^(s/2) mod 4, where d = 1 requires s ≡ 0 mod 8. This settles the case left open by Shinohara
in 1971. The note is unrefereed.

## Statement checked
- **Problem list.** 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, arXiv:math/0406190. Problem 12.21 (p. 540), attributed to
  Y. Shinohara: "If n=4k+1 with k>0, is there a knot with determinant n and signature 4?"
  - The remark by A. Stoimenow (p. 541): the form 4k + 1 comes from Murasugi's congruence and k ≠ 0 from the
    signature theorem for even unimodular forms; every prime divisor of a counterexample n > 1 is ≡ 1 mod 24 and at
    least 2857; values of certain elementary symmetric polynomials at positive odd arguments are never
    counterexamples, but a proof along these lines appears number-theoretically hard.
- **Origin.** Y. Shinohara, On the signature of knots and links, Trans. AMS 156 (1971) 273–285,
  doi:10.1090/S0002-9947-1971-0275415-1 (read in the AMS scan).
  - p. 278, (3.7): |Δ(−1)| ≡ (−1)^m mod 4 iff |σ| ≡ 2m mod 4, and |Δ(−1)| = 1 implies σ ≡ 0 mod 8 (Corollary 5 is
    credited there to Murasugi, Theorem 5.6).
  - Theorem 6: knots with (|Δ(−1)|, |σ|) = (4m+1, 8n), (8m+5, 8n+4), (4m+3, 4n+2) for all m, n ≥ 0; the second
    family is T(2, 8m+5) # copies of T(3,5) or its mirror image.
  - Remark after Theorem 6: (8m+1, 8n+4), m > 0, is stated to be realizable for m ≡ ±1 mod 3 (no proof given);
    the case m ≡ 0 mod 3, i.e. det ≡ 1 mod 24, is stated to remain open, with an affirmative answer expected.
- **Asked again.** A. Stoimenow, Determinants of knots and Diophantine equations, Acta Arith. 129 (2007) 363–387,
  doi:10.4064/aa129-4-6 (read in the publisher's open-access PDF), §5.1:
  - Question 5.1: is S := {det(K) : σ(K) = 4} equal to 5 + 4N?
  - Remark 5.1: other signatures ≡ 4 mod 8 reduce to σ = 4 by connected sums with 10_124 = T(3,5) and its mirror;
    prime examples follow from composite ones by Bleiler's method.
  - Proposition 5.1: S contains every 4l + 1 with a divisor 4k + 3; S is closed under multiplication by 4k + 1;
    S contains the progressions 5 + 8k, 5 + 12k, 9 + 12k and all 4l + 1 in (1, 2209); 1 ∉ S (Theorem 1.1).
  - Corollary 5.1: every prime divisor of a number 4l + 1 ∉ S is ≡ 1 mod 24 and at least 33049; a search over the
    primes up to 4·10^9 shows that S contains every 4l + 1 ≥ 5 up to that bound.
  - It also asks whether ±1 plays a special role in Theorem 1.1 (no knot with det 1 and σ ≡ 4 mod 8).
- **Related.** A. Stoimenow, Polynomial values, the linking form and unknotting numbers, Math. Res. Lett. 11 (2004)
  755–769, arXiv:math/0405076 (arXiv source read): if det K is a square with no prime divisor ≡ 3 mod 4 and
  σ(K) = 4, then u(K) > 2; by computer, every non-square 4k + 1 < 400 is the determinant of a knot with u = 2 and
  σ = 4.
- **Corpus record.** ulamai/UnsolvedMath version 1.6.0 (unchanged at commit 372682f), AMR-103-0226, status `open`,
  research classification OPEN-TRIAGE. The statement field agrees with the problem list. The title field is
  truncated ("Problem 12.21 — (Y.").

## Readings
| Reading | Answered? | Witness |
|---|---|---|
| σ(K) = +4 (as printed) | yes, for every n = 4k + 1, k > 0 | Theorem 1.1 |
| σ(K) = −4 (other sign convention) | yes | mirror images |
| Shinohara's form: \|Δ(−1)\| = 8m + 1 (m > 0), \|σ\| = 8j + 4 for all j ≥ 0 | yes | Corollary 1.3 (connected sums with a knot of det 1, σ = 8, or its mirror) |
| Stoimenow's Question 5.1: S = 5 + 4N? | yes | Theorem 1.1 with Lemma 2.1 |
| Stoimenow's side question: is ±1 special in "no knot with det 1 and σ ≡ 4 mod 8"? | yes: apart from the values excluded by Murasugi's congruence, 1 is the only exception | Theorem 1.2 |
| prime (or hyperbolic) knots required | yes | Remark 5.5 (Bleiler's method via Stoimenow's Remark 5.1; Stoimenow, IMRN 2008) |
| n = 1 (k = 0) | excluded in the problem, and impossible | Lemma 2.1 |

## Results in the paper
- **Lemma 2.1 (Murasugi).** det(V + V^T) ≡ (−1)^g mod 4 for every 2g × 2g V with det(V − V^T) = 1; hence
  det K ≡ (−1)^(σ/2) mod 4, and det K = 1 forces σ ≡ 0 mod 8. A short self-contained proof is given; credited to
  Murasugi (1965, Thm 5.6), Shinohara (1971, (3.7)) and Stoimenow (2007, Thm 1.1).
- **Theorem 2.2 (Seifert 1935)** and **Lemma 2.3** (every even matrix of odd determinant is V + V^T with
  det(V − V^T) = 1).
- **Lemma 3.1.** G(b,c,y) = [[1,1,0,0],[0,b,y/2,0],[0,y/2,c,1],[0,0,0,1]] (y even) has G − G^T = J ⊕ J and
  det(G + G^T) = (4b−1)(4c−1) − 4y^2; for b ≥ 1 and positive determinant the form is positive definite (σ = 4).
- **Lemma 3.2.** For every non-square n ≡ 1 mod 4 there are a prime l ≡ 3 mod 4, l ∤ n, and an even y with
  l | n + 4y^2 (CRT, Dirichlet's theorem, Jacobi reciprocity).
- **Theorem 1.1.** Cases: (a) a prime factor l ≡ 3 mod 4: G((l+1)/4, (n/l+1)/4, 0) (Stoimenow, Prop. 5.1(1));
  (b) n not a square: G from Lemma 3.2 (new); (c) n = s^2 with all prime factors of s ≡ 1 mod 4: G_p ⊕ T(n/p),
  genus 3 (Stoimenow, Prop. 5.1(2)).
- **Theorem 1.2 / Corollary 1.3.** Full classification of (det, σ); realization by T(d), E(d), its mirror, and
  Theorem 1.1, shifted by connected sums with a knot realizing the E8 Seifert matrix V_8 (det 1, σ = 8) or its
  mirror; Shinohara's cases (1), (3) and his open case (4).
- **Remark 3.3.** The prime l = 3 gives the progressions 9 + 12k and 5 + 12k (Stoimenow); with Shinohara's 5 mod 8
  this covers n ≢ 1 mod 24. For primes p ≡ 1 mod 4 below 10^7 the least admissible l is at most 139 (computation).
  Examples: G(3,71,8) for 2857 and G(5,437,6) for 33049.
- **Proposition 5.1 and Remark 5.2.** No G(b,c,y) (b ≥ 1) has a square determinant whose prime factors are all
  ≡ 1 mod 4 (the congruence of Stoimenow 2004). Knots with σ = 4 and square determinant have non-cyclic
  H_1 of the double branched cover (Jabuka 2012 / Milgram), so they are never two-bridge; this confirms the first
  part of a conjecture at the end of the arXiv version of Stoimenow 2004. Two-bridge examples exist for every
  non-square n ≤ 20001 (computation); genus-2 examples for every square s^2, s ≤ 101, all of whose prime factors
  are ≡ 1 mod 4 (computation; s = 97 and 101 were found in the second verification run).
- **Remark 5.3 (sketch, not used).** A lattice-theoretic route (Nikulin 1979, Thm 1.10.1; Wall 1963; Milgram's
  formula) together with Lemma 2.3 and Seifert's theorem; it would give genus 2 in all cases.
- **Remark 5.4, Table 1.** Table knots for n ≤ 49 (5_1, 3_1#3_1, 7_3, 7_5/8_2, 9_4, 5_1#4_1, 9_7, 9_10/9_11,
  9_13, 9_18/9_20, 9_23, 5_2#5_2); S(2857,652) (19 crossings) and S(33049,9229) (24 crossings), the least crossing
  numbers among two-bridge knots of these determinants with |σ| = 4 (computation).

## Computations (exact; scripts and outputs in reproducibility/)
- **Lead** (`lead/verify_note.py`, standard library, about 15 s, 3,115,300 checks, all passed):
  - Lemma 3.1 and the blocks E(u), T(m) proved as polynomial identities;
  - Lemma 3.2 literally for all 249,501 non-squares n ≡ 1 mod 4, 5 ≤ n ≤ 1,000,001 (largest Dirichlet prime
    287,024,779 at n = 889,681);
  - Theorem 1.1 for every n ≡ 1 mod 4, 5 ≤ n ≤ 1,000,001 (cases a/b/c = 162,118/87,760/122);
  - Proposition 5.1 sanity check; V_8 and all 530 admissible pairs (d, s), d ≤ 101, |s| ≤ 20 (Theorem 1.2);
  - Table 1 (with calibration on 3_1–7_7), S(2857,652), the examples for 2857 and 33049, and the genus-2 matrices
    for squares.
- **Finder** (`finder/`): the construction for every n ≤ 2·10^6; Murasugi's congruence on 4000 random Seifert
  matrices; least auxiliary primes for p < 10^7; enumeration of two-bridge knots up to 12 crossings (counts agree
  with Ernst–Sumners); two-bridge scan up to 20001; genus-2 square search; named table knots.
- **First independent verification run** (`independent/`, AI-assisted, own code): Lemma 3.1 by exact evaluation on a grid
  exceeding the multidegree; Murasugi's congruence on 3000 random Seifert matrices with a general unimodular skew
  part (and σ ≡ 0 mod 8 in all 2423 cases with |det| = 1); the proof followed literally for every n ≤ 10^6 + 1
  (same case counts and largest prime as the lead); two-bridge determinants and signatures by three methods,
  agreeing on all 8,282 pairs (p, q) with p ≤ 201.
- **Second independent verification run** (`independent_run_2/`, AI-assisted, code written from scratch before
  any packaged script was read):
  - the proof followed literally for every n ≡ 1 mod 4, 5 ≤ n ≤ 1,000,001, with every symbol identity of
    Lemma 3.2 asserted and an exact inertia computation for every matrix: cases a/b/c = 162,118/87,760/122 and
    largest Dirichlet prime 287,024,779 (n = 889,681), the same values as the lead;
  - determinants and signatures from knot DIAGRAMS (Fox colouring matrix; Goeritz matrices and the
    Gordon–Litherland formula for both checkerboard colourings), calibrated on all 170 four-plat knot diagrams
    with at most 10 crossings: every knot of Table 1 has determinant n and |σ| = 4; S(2857,652) (19 crossings) and
    S(33049,9229) (24 crossings) have determinant p and |σ| = 4, and no two-bridge knot of determinant 2857
    (resp. 33049) with |σ| = 4 has fewer crossings;
  - Lemma 3.1 identities; Lemma 2.1 on 2000 random Seifert matrices; the proof of Lemma 2.3 implemented and
    tested on 600 random matrices; V_8 and all 530 pairs of Theorem 1.2; Remark 3.3 (least l ≤ 139 for all
    332,180 primes p ≡ 1 mod 4 below 10^7; G(3,71,8), G(5,437,6)); Proposition 5.1 sanity check;
  - genus-2 Seifert matrices of determinant s^2 and signature 4 for every s ≤ 101 all of whose prime factors are
    ≡ 1 mod 4 (including s = 97, 101), all with non-cyclic cokernel as Remark 5.2 predicts; every non-square
    n ≤ 20001 is the determinant of a two-bridge knot with |σ| = 4, and every two-bridge knot S(s^2, q) with
    s^2 ≤ 5041 has σ ≡ 0 mod 8.
- All lead, finder and first-run outputs were regenerated from the packaged scripts (in the second run, from an
  extracted copy of the source archive) and agree with the recorded ones apart from timing lines.

## Independent verification runs

### First run
Verdict of the run (2026-10-02): **correct; answers the question as posed and as intended; no prior answer found.**

| Item | Verdict |
|---|---|
| Statement fidelity (problem list, Shinohara 1971, Stoimenow 2007 via zbMATH) | CONFIRMED |
| Proofs (Lemma 2.1, Seifert realization, Lemma 2.3, Lemma 3.1, Lemma 3.2, assembly) | CONFIRMED, no mathematical error |
| Computations | CONFIRMED (reruns byte-identical; own code up to 10^6) |
| Answer as posed | CONFIRMED (yes) |
| Novelty | no published answer found; lattice-theoretic alternative noted; MathSciNet and Google Scholar not checked |
| Presentation | required fixes in history and attribution |

All required fixes were applied:
1. Shinohara 1971 (Theorem 6 and the remark after it, pp. 278–279) is cited as the origin; the known cases
   (det ≡ 5 mod 8; det ≢ 1 mod 24) are stated, and the genuinely new case det ≡ 1 mod 24 is identified.
2. The theorem is also stated in Shinohara's form (all |σ| = 8j + 4) together with the complete classification
   of pairs (det, σ) (Theorem 1.2, Corollary 1.3), which also answers Stoimenow's question about ±1.
3. Stoimenow, Math. Res. Lett. 11 (2004) is cited and related to the construction (same congruence; computer
   evidence for non-squares below 400).
4. The lattice-theoretic alternative (Nikulin, Wall, Milgram) is mentioned and the novelty claim is limited to
   "no published answer found".
5. The exclusion of det 1 with σ ≡ 4 mod 8 is credited to Murasugi's congruence, Shinohara (1971) and Stoimenow
   (2007, Theorem 1.1).
6. Optional items: prime/hyperbolic realizations (Bleiler via Stoimenow 2007; Stoimenow 2008), explicit two-bridge
   knots for non-squares up to 20001, and the truncated title of the dataset record are mentioned.

In addition, the lead read Stoimenow 2007 (Sections 1, 2 and 5) in the open-access PDF and added what it already
contains: Proposition 5.1 (cases (a) and (c) of the proof), Corollary 5.1 (bound 33049) and the computer search up
to 4·10^9.

### Second run
Verdict of the run (2026-10-02/03, on the revised note): **correct; answers the question as posed and as intended;
no prior answer found; no mathematical error.**

| Item | Verdict |
|---|---|
| Statement fidelity (problem list pp. 540–541, published PDF) | CONFIRMED |
| Credit and history, against the originals: Shinohara 1971 pp. 278–279 (Cor. 5, (3.7), Thm 6, Remark), Murasugi 1965 Thm 5.6, Stoimenow 2007 (Thm 1.1, §5.1: Question 5.1, Remark 5.1, Prop. 5.1, Cor. 5.1, search to 4·10^9, the ±1 question), Stoimenow 2004 (theorem, computer remark, final conjecture), Jabuka 2012, Stoimenow 2008 (zbMATH review) | CONFIRMED; one precision fix (what exactly was open) |
| Proofs (Lemma 2.1, Theorem 2.2 and mirror images, Lemma 2.3, Lemma 3.1, Lemma 3.2 with all edge cases, Theorem 1.1, Theorem 1.2 with the E8 matrix, Corollary 1.3, Proposition 5.1, Remarks 3.3, 5.2, 5.3) | CONFIRMED line by line |
| Computations | CONFIRMED with own code, including a diagram-based method for the knots; all packaged scripts rerun |
| Novelty | no published answer found (citers of Shinohara 1971, Stoimenow 2004 and 2007 searched again) |
| Presentation | conforms to the house style; minor fixes |

Required fixes of the second run, all applied:
1. Precise statement of what was open and what is new: by Shinohara and Stoimenow (cases 5 mod 8 and ≢ 1 mod 24,
   a prime factor ≡ 3 mod 4, multiplicativity, Cor. 5.1, search to 4·10^9) the question had been reduced to primes
   p ≡ 1 mod 24; the new part is Lemma 3.2 with the matrices of Lemma 3.1, which settles every non-square and hence
   this case, and the genus bound (introduction and "Scope and priority").
2. Abstract: the fact that determinant 1 forces σ ≡ 0 mod 8 is attributed to the even unimodular symmetrized
   Seifert form, not to Murasugi's congruence.
3. The second run is recorded in the note (Verification paragraph, with a reference to Gordon–Litherland for the
   diagram-based signature), in this report and in `reproducibility/independent_run_2/`.
4. Bibliography: the dataset reference gives the corpus version (1.6.0); the record is unchanged in the current
   snapshot.

Minor changes: "Shinohara considered whether" instead of "asked whether"; Remark 3.3(a) reworded; Remark 5.2 now
lists genus-2 examples for every s ≤ 101 (adding 97 and 101).

## Relation to the literature, novelty and scope
- **Searches (2026-10-02).** arXiv API (several keyword queries), Crossref, OpenAlex (works citing Shinohara 1971:
  36; citing Stoimenow 2007: 1, Jabuka arXiv:1204.4965), Semantic Scholar (citers of the problem list: 128; of
  Stoimenow 2007: 2), zbMATH (Stoimenow's 113 items; review Zbl 1149.11016), and two general web searches.
  - Jabuka's paper (read) proves σ ≡ 0 mod 8 for vanishing Witt class of the linking form; it does not address
    S = 5 + 4N.
  - O. A. Ivanov, Zap. LOMI 83 (1979) (citing Shinohara; read): no existence result.
  - Borodzik–Friedl, Livingston, Conway's survey of Levine–Tristram signatures: not relevant.
- **Searches of the second run (2026-10-02/03).** OpenAlex and Semantic Scholar citers of Shinohara 1971 (36/44),
  Stoimenow 2007 (1/2) and Stoimenow 2004 (37/41); six arXiv API keyword queries; zbMATH API queries; one general
  web search (three in total).
  - Gros–Pastor–Ramírez Alfonsín, arXiv:2409.14133 (determinants via Fourier–Hadamard transforms): not relevant.
  - Jost–Lewark, arXiv:2601.16588 (2026; source read): evaluations of the Jones and Q polynomials from the double
    branched cover and a counterexample to another conjecture of Stoimenow 2004; nothing on knots of signature 4.
  - Stoimenow, Some examples related to knot sliceness (arXiv:math/0412276, source read) and Graphs, determinants of
    knots and hyperbolic volume (PDF read): not about this question.
- **Caveats.** MathSciNet and Google Scholar were not searched. The result also follows from the classical existence
  theory of even lattices with prescribed discriminant form (Remark 5.3), so it may be known to experts. This
  negative search is not a proof of priority.
- **Scope.** The note answers Problem 12.21 as stated, Stoimenow's Question 5.1, and Shinohara's open case. The
  necessary conditions, the cases det ≡ 5 mod 8 and det ≢ 1 mod 24, the case of a prime factor ≡ 3 mod 4, the
  reduction of squares (multiplicativity) and the bound 33049 are due to Murasugi, Shinohara and Stoimenow; they
  reduce the question to primes p ≡ 1 mod 24. The new part is Lemma 3.2 with the genus-2 matrices of Lemma 3.1,
  which settles every non-square n ≡ 1 mod 4 and hence that case, together with the genus bound. Whether genus 2
  suffices for squares all of whose prime factors are ≡ 1 mod 4 is only sketched (Remark 5.3) and checked by
  computer for s ≤ 101.

## 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/