# Verification report — AMR-103-0016 (Ohtsuki's list, Problem 1.16 of E. Ferrand and A. Stoimenow: signature, HOMFLY and Kauffman polynomial of alternating links) Verification date: 2026-10-10. **Verdict.** The problem of the record is **answered in print by others**; the note does not solve it. What the note proves, with complete proofs, is a sharper form of the first inequality and two applications. - **The record is out of date.** Problem 1.16 asks whether σ(L) ≥ min deg_l P_L(l,m) ≥ min deg_a F_L(a^-1,z) for every alternating link L. The second inequality is Rutherford (Int. Math. Res. Not. 2006, Art. ID 78591), Corollary 5.4, with a construction of Ng (Algebr. Geom. Topol. 5 (2005), Section 4; quoted by Rutherford as Proposition 11); it also follows from Tanaka (Topology Appl. 153 (2006)). The first inequality is Ito (Bull. Aust. Math. Soc. 114 (2026) 182–190, published online 2 March 2026; arXiv:2504.13491, Theorem 5 of the arXiv version). The answer is yes. - **Proved in the note (Theorem 1.1).** For a connected reduced alternating diagram D of L, with d_-(D) = (n_- + s − s_A)/2: w − s + 1 ≤ min deg_l P_L ≤ σ(L) − d_-(D) = w − s + 1 + d_-(D); d_-(D) is a non-negative integer, and d_-(D) ≥ 1 if D has a negative crossing. Hence σ(L) = min deg_l P_L if and only if D is positive, and min deg_l P_L ≤ σ(L) − 2 otherwise. Corollary 1.2 is the version for disconnected diagrams (it contains Ito's Theorem 5). - **Consequences (Corollary 1.4).** A positive answer to Question 8 of Orevkov (arXiv:2007.00093): the inequality 2r^-(D) ≤ d^-(D) holds for every connected reduced alternating diagram. Part (i) of Conjecture 1 of Ito's paper for alternating links; part (ii) for reduced alternating diagrams only. - **Derivable from statements in print (Section 6.4 of the note; found by the third verification run).** The case of equality of Theorem 1.1, the bound min deg_l P_L ≤ σ(L) − 2 for diagrams which are not positive, and Corollary 1.4(b), (c) follow from Ito's theorem on the coefficient of the highest power of m (Cromwell's theorem; Theorems 7 and 8 of the arXiv version) together with the inequality min deg_l P ≤ min deg_l P_max − 2 for non-positive reduced homogeneous diagrams in the proof of Corollary 4.16 of A. Stoimenow, "Realizing strongly quasipositive links and Bennequin surfaces", Publ. RIMS 62 (2026) 245–339 (the argument of his Lemma 4.12: the substitution of Lickorish and Millett which makes P trivial, and Cromwell's Theorem 4). Neither paper states these consequences; the note does not claim them. - **Not a new theorem (Corollary 1.3).** Every reduced alternating diagram of a quasipositive alternating link is positive. This is the alternating case of Theorem 5.24 (homogeneous quasipositive links are positive) of the version of Stoimenow's paper on his home page (dated 11 March 2026; the section is not in the published version; its proof was not checked here). With Ito's Theorem 4 the statement also follows from the statements in print just mentioned. The note gives a short independent proof of the alternating case. - **Novelty.** The upper bound with d_-(D), which is stronger than σ − 2 when d_-(D) ≥ 3, and with it Corollary 1.4(a), were not found in the sources read. The proof is half a page from classical facts; it may be known to specialists. No priority is claimed. 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, arXiv:math/0406190; Problem 1.16 (E. Ferrand, A. Stoimenow), in Section 1.2. - Read in the arXiv source (twice: for the first version and by the third verification run): the definitions of V, P and F in Sections 1.1 and 1.2, the statement of the problem and the remark after it. - HOMFLY polynomial of the list: l^-1 P(L+) − l P(L-) = m P(L0), P(unknot) = 1. Kauffman polynomial: F = a^(-w(D)) [D], [D+] + [D-] = z([D0] + [D∞]), a kink of writhe +1 multiplies [D] by a. The list states V(t) = P(t, t^(1/2) − t^(-1/2)) = F(−t^(-3/4), t^(1/4) + t^(-1/4)). - The third run checked that the Kauffman polynomials computed with the kink convention of the note satisfy this identity on the eight diagrams of Table 1; with the other convention they would not. - The remark after the problem (by Stoimenow) attributes the second inequality to a conjecture of Ferrand and recalls that both min deg_l P and σ are at most 1 − χ(L). - The sign of σ is not fixed in the list. It is forced by the statement: with the other sign the positive trefoil would violate it. The convention σ(positive trefoil) = +2 is that of Ng (2005) and Ito (2026). - **Corpus record.** ulamai/UnsolvedMath, AMR-103-0016 (dataset version 1.6.0; upstream status `open`). Its statement is the statement of the list with flattened typography; its title is cut after "(E.". ## Readings | Reading | Answer | Where | |---|---|---| | The question as posed: both inequalities for every alternating link (all numbers of components, all orientations, split and composite links included) | yes; in print (Ito; Rutherford with Ng; Tanaka) | Section 1.1, Theorem 2.4, Corollary 1.2 | | First inequality, sharper form | min deg_l P_L ≤ σ(L) − d_-(D) for connected reduced alternating D; equality in σ ≥ min deg_l P iff D is positive | Theorem 1.1 | | The same, from statements in print | min deg_l P_L ≤ σ(L) − 2 for non-positive D (Ito with Stoimenow's published inequality) | Section 6.4 | | Split alternating links | min deg_l P_L ≤ σ(L) − (r − 1) − Σ d_-(D_i) for a reduced alternating diagram with r components | Corollary 1.2 | | Opposite sign of σ; l replaced by l^-1; F(a,z) in place of F(a^-1,z) | each of these readings fails on a trefoil | Table 1 and its caption | | Non-alternating links | not the question; both inequalities fail there (for instance 10_139: σ = 6, min deg_l P = 8) | Remark 3.7, Section 7 | | Diagrams which are not reduced | the inequality between link invariants is unaffected; the bound with d_-(D) fails (unknot with one negative kink) | Remark 3.7 | ## Results in the paper - **Section 2 (conventions).** Crossing signs, A- and B-smoothings, Seifert state; three elementary facts (S1)–(S3); Lemma 2.1 (faces of an alternating diagram: s_A + s_B = n + 2; reduced implies that the two arcs of the all-A state at a crossing lie on different circles; oriented smoothing keeps a diagram alternating); the HOMFLY polynomial and Theorem 2.2 (Morton; Franks–Williams; cited); the Jones polynomial through the bracket, formula (5); signature and nullity, Lemma 2.3 (disconnected Seifert surfaces); the Kauffman polynomial and the second inequality as a reported theorem (Theorem 2.4), with the translation of Rutherford's normalisation. Table 1: eight small diagrams. - **Section 3.** Lemma 3.1 (degree comparison under the Jones substitution); Lemma 3.2 (Morton's bound max deg_m P ≤ n − s + 1, with proof, and the parity of the exponents); Lemma 3.3 (Kauffman, Murasugi, Thistlethwaite, with proof); Lemma 3.4 (Traczyk, Theorem 2, formula (2); cited, with the translation of the sign convention); Lemma 3.5 (circle count). Proof of Theorem 1.1 and of Corollary 1.2. Proposition 3.6: d_- = s − s_- = s_+ − 1, every spanning tree of the Seifert graph has d_- negative edges, σ = w − s + 2 s_+ − 1. Remarks on the hypotheses and examples (Figure 2: figure-eight knot). - **Section 4.** Lemma 4.1: a band move changes exactly one of signature and nullity, by exactly one. Corollary 4.2: |σ| ≤ k − N + 1 + nullity for the closure of a product of k conjugates of generators in B_N. - **Section 5.** Proof of Corollary 1.3 from three inequalities: Morton–Franks–Williams for the closed braid, Corollary 1.2, Corollary 4.2. - **Section 6.** Orevkov's inequality 2r^- ≤ d^- (Corollary 1.4(a)); Ito's Theorems 2 and 5, his remark on 6_1, and his Conjecture 1 (Corollary 1.4(b), (c)); Stoimenow's Section 5 (Theorem 5.24, Corollaries 5.28–5.30); Section 6.4: the case of equality from statements in print, and the comparison of the bounds. ## Computations (sanity checks; programs and outputs in reproducibility/) No proof depends on a computation. All arithmetic is exact. Four independent sets of programs; in every test listed here the number of failures is 0. - **The exact program of the note** (`writing_stage/check_paper.py`, standard library, one second; TOTAL failures: 0): every number printed in Table 1, in the captions and in the examples (6_1, T(2,4), the kink), recomputed with two libraries. - **Programs with which the results were first obtained** (`original/`; HOMFLY by Regina 7.4.1 and by an own skein recursion, signature by an own Gordon–Litherland routine, own Kauffman polynomial): - conventions on the trefoils, the Hopf links and the figure-eight knot; three routes to the Jones polynomial on 801 knots and 1 268 links of the tables; - KnotInfo (12 965 knots with at most 13 crossings) and LinkInfo (4 188 oriented links with at most 11 crossings): signature, HOMFLY and Kauffman polynomial against the tabulated values; all statements on the 6 729 + 2 242 alternating entries and their mirror images (17 942 oriented links); - the census of Hoste and Thistlethwaite in SnapPy: all 68 448 prime alternating knots and links with at most 14 crossings, every orientation, both chiralities: 296 622 oriented links (148 311 computed diagrams, the mirror images by the mirror rules); - 3 182 diagrams of structured families up to 107 crossings (19 310 oriented links, 5 416 of them split) and 1 380 random alternating diagrams with 15 to 60 crossings (9 014 oriented links); - Lemmas 3.1, 3.2, 3.5 and Theorem 2.2 on all 17 153 table diagrams, Lemma 3.3 on the 13 892 A-adequate ones; - 1 222 random quasipositive braids with 6 536 band moves. - **Verification run A** (`verification_run_A/`; own programs, no knot software for invariants): - 27 small diagrams, with polynomials obtained directly from the skein relation; - validation of its tools: Goeritz form against Seifert form on 6 000 random braids; a signature computed without surfaces on 600 census diagrams; - all prime alternating knots and links with at most 12 crossings (19 834 oriented diagrams, both chiralities computed) and a sample of 2 400 diagrams with 13 crossings (10 294 oriented): 25 statements per diagram; the 1 054 positive diagrams are exactly those with min deg_l P = σ; - 686 connected sums, 321 split unions, 270 further alternating diagrams, 1 650 closed alternating braids up to 44 crossings; 600 non-reduced alternating diagrams (the bound of Theorem 1.1 fails for 183, as it may); 1 500 A-adequate diagrams; - Lemma 4.1 on 38 725 + 25 065 band moves; Traczyk's Theorem 1 on 9 242 smoothings; - 42 177 quasipositive braids with 193 593 band moves; 9 542 closures have the polynomial, signature and nullity of a census diagram, and all these census diagrams are positive; - the quasipositive braids listed in KnotInfo (156 alternating knots) and LinkInfo (227 alternating links): the alternating table diagram is positive (negative where the table lists the mirror image) in all cases; - own Kauffman polynomial on all 1 868 oriented census diagrams with at most 10 crossings. - **Verification run B** (`verification_run_B/`; own diagram and signature code, HOMFLY by Regina, second methods through Spherogram, Regina's second algorithm and an own Hecke-algebra routine): - the census up to 14 crossings: 296 622 oriented diagrams, each built and computed separately; up to 13 crossings (72 838 diagrams) every invariant by a second method; - 4 093 alternating table entries (8 186 diagrams with mirror images); 1 500 connected sums up to 24 crossings; 1 500 split unions; 8 118 band moves, with a second computation of the nullity; - 12 770 random quasipositive words with 57 570 band moves; the closures were simplified with Spherogram: 3 237 unlinks, 7 171 with a reduced alternating diagram (all positive), 2 063 certified not alternating, 299 undecided; a control on 3 100 alternating braid closures (2 159 recognised as non-positive); - a sensitivity test: seven deliberately wrong variants of the statements fail on 634 to 5 610 of 5 610 diagrams; - a re-run of the nine commands of `original/`: outputs identical up to the lines with running times. - **Verification run 3** (`independent_run_2/`; made on the final text; programs written from the statements of the note before the other programs of the package were opened; one library of its own, standard library only, for all invariants: HOMFLY by a skein tree and, for closed braids, by a Hecke algebra; Kauffman bracket by the state sum; Kauffman polynomial by its skein tree; signature and nullity by the Gordon–Litherland form for both chessboard colourings and, for closed braids, by a Seifert form derived for this run): - every number of Table 1 and of its caption (all eight polynomials P; e_F from its own Kauffman polynomial), Figure 2, Remark 3.8, Sections 2.2 and 6.2: 140 checks; - all 4 684 prime alternating knots and links with at most 12 crossings, all orientations, both chiralities, each of the 39 668 oriented diagrams computed separately: up to 45 statements per diagram (the hypotheses; (S1), (S3); Lemma 2.1(a), (b), (c); Lemmas 3.1–3.5; Theorem 2.2; Theorem 1.1(a)–(d) with the case of equality and the parity; Proposition 3.6, part (b) on random spanning trees); bracket by the state sum up to 10 crossings; equation (6) and the second inequality up to 9 crossings (1 236 diagrams). σ − min deg_l P takes the values 0, 2, …, 12; min deg_l P = σ − d_- only for the 1 606 positive diagrams; - alternating diagrams on the shadows of 720 random closed braids: 4 276 reduced oriented diagrams (3 832 with at least two components, up to 16 crossings); 3 824 diagrams with nugatory crossings (Traczyk's formula and Proposition 3.6(a) hold; the bound of Theorem 1.1 fails for 394, as it may; removal of the nugatory crossings by twists gives reduced alternating diagrams with the same P, σ and nullity); 296 connected sums; 296 split unions with up to two extra circles (Corollary 1.2 and the last sentence of Section 6.1); - 1 500 random braids (Hecke algebra = skein tree, Seifert form = Gordon–Litherland form); Lemma 4.1 on 6 000 + 2 804 band moves; 3 000 quasipositive words with 11 905 band moves (Lemma 4.1, Corollary 4.2, step (1) of Section 5); - quasipositive words whose closures are alternating (closed braid diagrams simplified with Spherogram; the polynomial, the signature and the nullity of the diagram found must be those of the closed braid): of 1 500 random words 1 196 have an alternating diagram (268 unlinks, 928 with crossings, 651 of them with at least two components); of the 1 100 words with two or three bands in B_3 and conjugating words of length at most one, 972 (66 unlinks, 906 with crossings). All 1 834 reduced alternating diagrams with crossings are positive, with σ = k − N + r and min deg_l P = k − N + 1. Control: among 400 closed alternating braids the same procedure finds 345 non-positive reduced alternating diagrams; - Section 6.4 on the 11 220 oriented census diagrams with at most 11 crossings: P(l, l^-1 − l) = 1; the coefficient of m^(n−s+1) contains l^σ; both inequalities of Stoimenow's proof; of the 10 586 non-positive diagrams, the bound of Theorem 1.1 (with the parity) equals the better bound from the literature for 7 750 and is stronger for 2 836; it is attained for 6 296, the literature bound for 4 656; - Orevkov's inequality in his own terms (braids of minimal index of KnotInfo) for the 563 alternating knots with at most 11 crossings; the 156 alternating knots with a quasipositive braid in KnotInfo: the alternating table diagram of the same chirality is positive. - **Re-runs.** `reproducibility/run_quick.sh` re-runs, in one process, every program of the first three sets which needs at most about five minutes (among them the complete census up to 14 crossings with the programs of run B and the complete census up to 12 crossings with the programs of run A) and compares the results with the recorded outputs. It was run on 2026-10-10 when the note was written (35 minutes; `reproducibility/RERUN_LOG.txt`) and again by the third verification run, from an extracted copy of the archive (36 minutes; `reproducibility/independent_run_2/outputs/rerun_of_run_quick.log`): all comparisons succeeded both times. Two comparisons are made up to documented differences: the table look-up of run A (the dictionary is rebuilt without the 13-crossing sample) and the split between "alternating" and "undecided" in `qp_test.py` of run B (random simplification moves; recorded 7 171 and 299; in the re-runs 7 177 and 293, 7 174 and 296; all alternating closures positive). The programs of run 3 were run again from their final texts to produce the recorded outputs, and once more with `independent_run_2/programs/run_all.sh` from an extracted copy of the final archive (11 minutes with five processes; all comparisons succeeded). Details are in `reproducibility/README.md` and `reproducibility/independent_run_2/README.md`. ## Independent verification runs The results were first obtained with proofs of Theorem 1.1, Lemma 4.1 and Corollary 1.3 (there: Theorem A, Lemma 6, Corollary C). Three independent verification runs, all AI-assisted, were made on 2026-10-10; each wrote its own programs. Runs A and B examined the first written version of the results and did not see each other: run A every statement, every proof and all conventions; run B the literature, the credits and the novelty, and it repeated the computations. Run 3 examined the final text of the note. | Item | Run A | Run B | Run 3 (final text) | |---|---|---|---| | Statement of the problem and conventions against the source | CONFIRMED | CONFIRMED | CONFIRMED (also the identity V = F(−t^(-3/4), t^(1/4) + t^(-1/4)) of the list) | | The record is answered in print (Ito; Rutherford, Ng; Tanaka) | CONFIRMED | CONFIRMED | CONFIRMED (sources read again) | | Cited theorems: Morton's bound, Morton–Franks–Williams, Kauffman–Murasugi–Thistlethwaite, Traczyk, non-vanishing determinant | CONFIRMED_WITH_FIXES (citations; Morton's bound and the degree formula re-proved, Traczyk's paper read, Morton–Franks–Williams re-proved for closed braids) | used as cited; Traczyk's formula re-derived from the Gordon–Litherland theorem | CONFIRMED (Traczyk's paper read in the publisher's file; his formula derived again from the Gordon–Litherland theorem, for all connected alternating diagrams) | | Theorem 1.1 with its lemmas, the split case, the comparison with Ito | CONFIRMED | CONFIRMED (re-derived); novelty: CONFIRMED_WITH_FIXES (credits) | CONFIRMED line by line in the final text, with Proposition 3.6 and the parity as now proved; novelty: CONFIRMED_WITH_FIXES (the case of equality follows from statements in print) | | Lemma 4.1, Corollary 4.2 | CONFIRMED | CONFIRMED | CONFIRMED | | Corollary 1.3: mathematics | CONFIRMED_WITH_FIXES (wording of the split case) | CONFIRMED | CONFIRMED | | Corollary 1.3: novelty of the statement | not its part | REFUTED: the statement is Theorem 5.24 of Stoimenow's home page text; the note presents it accordingly | not new: also a consequence of statements in print (Section 6.4) | | Corollary 1.4 | — | found the relation to Orevkov's question | (a) CONFIRMED against Orevkov's definitions; (b), (c) CONFIRMED, and derivable from statements in print | | Computations | own programs, 0 failures | own programs, 0 failures; original programs re-run | own programs, 0 failures; `run_quick.sh` re-run from the archive | No run found a counterexample, a wrong statement or a gap in a proof. **Corrections required by runs A and B**, all applied in the note: 1. (Run A) Morton's bound with a proof, for all diagrams: Lemma 3.2. 2. (Run A) Traczyk's theorem cited from the paper itself (Theorem 2, formula (2), for connected reduced alternating diagrams of oriented links, opposite sign convention; formula (1) is misprinted): Lemma 3.4 and the paragraph after it. 3. (Run A) Where "reduced" enters, with the example of the negative kink: Remark 3.7, Table 1. 4. (Run A) The split case with r defined as the number of components of the diagram: Corollary 1.2, Section 5. 5. (Run A, run B) Ito's Conjecture 1: part (i) for alternating links, part (ii) for reduced alternating diagrams only; "reduced" is needed in (ii): Corollary 1.4(b), (c), Section 6.2. 6. (Run A) A direct proof of d_- = s_+ − 1: Proposition 3.6. 7. (Run A) References for Lemma 4.1 (Murasugi 1965, Lemma 7.1, as recalled by Moore and Vazquez) and the two uses of Morton–Franks–Williams (general diagrams in Theorem 1.1, closed braids in Sections 5 and 6): Sections 2.2 and 4. 8. (Run A) The parity sharpening min deg_l P ≤ σ − 2 for non-positive diagrams: Theorem 1.1(c). 9. (Run B) Attribution of the question: Baader's paper on track knots for alternating knots; Boileau, Boyer and Gordon, Question 7.5 with Corollary 7.3: Section 1.3. 10. (Run B) Orevkov's Theorem 6 and Question 8, and the fact that Theorem 1.1 with Morton–Franks–Williams is exactly his inequality: Corollary 1.4(a), Section 6.1. 11. (Run B) Prior art for Theorem 1.1 in Stoimenow's text (Corollaries 5.28–5.30): Section 6.3. 12. (Run B) Second-hand citations checked against the sources or marked as such: Traczyk read; Morton's paper replaced by a proof (its content confirmed through the zbMATH review); Murasugi's lemma cited through Moore and Vazquez; the theorem number of Ito's journal version marked as unchecked: "Scope and priority". 13. (Run B) Corollary 1.3 presented as a short independent proof of the alternating case of Stoimenow's theorem, not as a new theorem: title of the corollary, abstract, Sections 1.3, 1.4, 6.3. 14. (Run B) What was read and what was not: "Scope and priority". **Changes made when the note was written, after runs A and B** (examined by run 3: all correct). - Proposition 3.6 is proved without the theorem that alternating diagrams are homogeneous: d_- = s − s_- comes from Lemma 2.1 applied to the diagram with all positive crossings smoothed, and s_+ + s_- = s + 1 from d_+ + d_- = s − 1. The statement on spanning trees then follows by counting. - The parity in Theorem 1.1(c) is derived from Traczyk's formula and Lemma 3.5 (σ ≡ μ − 1 mod 2) instead of the nullity. Lemma 3.2 also records the parity of the exponents of P. - Two remarks were added: in the proof of Corollary 1.2, that every component of a reduced diagram is itself reduced; at the end of Section 6.1, that Orevkov's inequalities also hold for reduced alternating diagrams with several components. **Corrections required by run 3**, all applied in the note: 1. (Credit; the main one.) The case of equality of Theorem 1.1(c), the bound σ − 2 for non-positive diagrams and Corollary 1.4(b), (c) follow from statements in print (Ito's theorem with an inequality in the proof of Corollary 4.16 of Stoimenow's published paper), and so does the statement of Corollary 1.3. The version examined by run 3 listed the case of equality as not found elsewhere. Now: abstract, Sections 1.2, 1.4, the new Section 6.4 (with the argument and a comparison of the bounds), "Scope and priority". 2. (Credit.) According to Stoimenow's home page text, σ ≥ min deg_l P with its case of equality follows for closures of alternating braids and for two-bridge links from explicit formulas of his paper of 2002 and of Murasugi: Section 6.3. 3. (Citation.) Rutherford's bounds are written for the Thurston–Bennequin number alone, in his notation; Ng's statement is in his Section 4 and is quoted by Rutherford as Proposition 11: Section 2.5. 4. (Citation.) Orevkov's Theorem 2 is stated for diagrams whose number of Seifert circles equals the braid index; the condition on single crossings is the theorem of Diao, Hetyei and Liu quoted there: Section 1.3. 5. (Bibliography.) Ito's article: Bull. Aust. Math. Soc. 114 (2026), no. 1, 182–190 (publisher's page; the journal version is not openly accessible); Lickorish–Millett added for the substitution. 6. (Citation of an unpublished text.) The numbering of the statements of the home page text is that of the version of 11 March 2026 and may change: bibliography entry. 7. (Text.) Remark 3.7, the Verification paragraph and "Scope and priority" brought to the final state (three runs; what was read; the searches). 8. (Wording.) Proof of Lemma 2.3: the meridian circle of the tube and its push-offs. 9. (Tests.) The programs and numbers of run 3 added to Section 7 and to the package. ## Relation to the literature, novelty and scope - **Searches (10 October 2026).** For the first version: arXiv, Crossref, OpenAlex, zbMATH, two web searches. Run A: two web searches. Run B: 31 arXiv queries (among them quasipositive with alternating, with homogeneous, with signature, with HOMFLY; alternating with HOMFLY and signature), the lists of citing works in OpenAlex and Semantic Scholar for Orevkov (2020), Ito (2022, 2026), Boileau–Boyer–Gordon, Baader's two papers and Stoimenow (2002, 2026), zbMATH, Crossref, three web searches. When the note was written: Crossref (33 records), zbMATH (4), arXiv (metadata of 13 e-prints); no web search. Run 3: 15 arXiv queries, the lists of citing works in OpenAlex for Orevkov (2020), Ito (2022, 2026), Boileau–Boyer–Gordon, Baader (2005), Stoimenow (2002, 2026) and Traczyk (2004), Semantic Scholar for three papers, zbMATH (3 queries), Crossref (all 24 DOIs of the bibliography and three further records), the preprint archive of the Max Planck Institute for Mathematics (3 queries), one web search. - No proof, announcement or refutation of "quasipositive alternating links are positive" was found other than Stoimenow's home page text. Ito's paper (arXiv 2025; journal 2026) writes that the question remains open even for alternating links. - No statement of the bound min deg_l P ≤ σ − d_-(D) was found. Nearest: the consequences of statements in print described above (min deg_l P ≤ σ − 2 for non-positive diagrams; min deg_l P ≤ n_+ − s − 1 in Stoimenow's proof); Ito's Conjecture 1 (equality case as a conjecture); Stoimenow's Corollary 5.29 on his home page (equality case for "ind-optimal" diagrams, proved with the τ-invariant). - **What was read.** When the note was written: Ito (arXiv:2504.13491v1) and Ito (arXiv:2007.03962v2) completely; Orevkov (arXiv:2007.00093v1) completely; Traczyk (2004) completely, in the publisher's open file; Baader (arXiv:math/0504594v1): the list of questions closing Section 4; Stoimenow's home page text (PDF of 78 pages fetched on 2026-10-10): the note on the home page, the statements of Sections 5.3–5.5 (Question 5.14, Theorem 5.24 with the beginning and the end of its proof, Corollaries 5.28–5.30); the arXiv sources of Ohtsuki's list, Rutherford, Ng and Ferrand (the statements quoted); Moore–Vazquez (arXiv:1806.02440v3): the sentence on Murasugi's Lemma 7.1; the zbMATH review of Morton (1986). By runs A and B in addition: Boileau–Boyer–Gordon (arXiv:1710.07658v2, Section 7); Ozbagci (arXiv:2307.07179v2); the published version of Stoimenow's paper (it has no section on quasipositivity of homogeneous links); the abstract of Baader's "Quasipositivity and homogeneity"; the zbMATH review of Stoimenow (2002). By run 3, all fetched again: the two papers of Ito, Orevkov and Traczyk completely; Section 7 of Boileau–Boyer–Gordon; Baader's list; of Stoimenow's home page text the Introduction, Section 2.3 and Sections 5.2–5.5 with the proof of Theorem 5.24; of the published version (open access, 95 pages) the contents, the passage of the Introduction which moves the applications to the four-genus and to quasipositivity elsewhere, and Section 4.4.1 up to Corollary 4.16 with the proofs; of Ohtsuki's list Sections 1.1 and 1.2; of Rutherford Sections 1, 2.1, 4.1 and 5; of Ng Theorem 4 with its proof and the remark after it; of Ferrand the Introduction, Section 3 and the question; the sentence of Moore–Vazquez; the abstract and a keyword scan of Stoimenow's newer home page text "Grid diagrams, link indices, and quasipositivity" (29 September 2026; on Whitehead doubles, not on this question). - **Not read.** Stoimenow (2002): no openly accessible copy was found (not among the papers on the author's home page, not on arXiv, not in the preprint archive of the Max Planck Institute for Mathematics; the publisher's page is not openly accessible). Tanaka (2006), Morton (1986), Franks–Williams (1987), Kauffman (1987), Murasugi (1987), Thistlethwaite (1987), Gordon–Litherland (1978), Murasugi (1965), Cromwell (1989), Orevkov (2002), Lickorish–Millett (1987); the journal versions of the papers read on arXiv (that of Ito's paper is not openly accessible); the papers mentioned only through Orevkov, Ito and Stoimenow (Murasugi–Przytycki, Cavallo–Collari, Hayden, Dynnikov–Prasolov, Yamada, Nakamura, Rudolph, Diao–Ernst–Hetyei–Liu). The proofs in the sources read were not checked; in particular the proofs of Stoimenow's Theorem 5.24, Lemma 4.12 and Corollary 4.16 were not checked (the inequalities quoted from them were tested on the census, see above). - **Caveats.** The theorem number of Ito's result in the journal version may differ from the arXiv version. Stoimenow (2002), where the chain of Problem 1.16 was first studied, could not be read; it cannot be excluded that the bound of Theorem 1.1 is contained in it or follows from it at once. A search which finds nothing is not a proof of novelty. - **Scope.** The note does not solve Problem 1.16 and claims no priority. It contributes a short proof of the first inequality in a sharper form (the bound with d_-(D)), its consequence for Orevkov's question, a second proof of the case of equality and of the two parts of Ito's conjecture for alternating diagrams, and a short independent proof of the alternating case of a statement announced by Stoimenow. ## 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/