{"id":"115cbb34-d9d2-4238-946b-abac48e2a853","arxiv_id":"2601.02323","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":3.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The product of characteristic polynomials of crossing matrices of the components of a braid system is invariant under Hurwitz equivalence, and its essential eigenvalues detect Euler fusion/fission.","lead":"This paper defines a polynomial built from the crossing matrix of a braid, and shows it stays the same when a tuple of braids is changed by the Hurwitz action. The authors use it to detect when two surface-link descriptions must be connected by an Euler fusion or fission.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Central invariant rests on Proposition 3 from the authors' own preprint [9]; if that conjugacy-invariance statement fails, Theorem 1 and Corollary 2 collapse.","rationale":"I read the paper as a short reduction: Hurwitz equivalence preserves the multiset of conjugacy classes, so any conjugacy invariant of individual braids gives a Hurwitz invariant by taking products or multisets. The formal logic from Proposition 3 to Theorem 1 and Theorem 2 is sound. I checked the key computations: Example 9's products match the displayed polynomials, and in Example 10 I independently computed P(σ3^{-1}σ2σ1^{-1}) = (x+1)^3(x-3), so the asserted essential eigenvalue sets are correct. The stabilization argument is also correct, aside from a harmless exponent typo (x^{m-2} should be x^{m-1} when σ_m is regarded in B_{m+1}); it does not affect E(b). The only genuinely load-bearing unsecured point is Proposition 3, quoted from the authors' own preprint [9]. This is not an internal inconsistency, but it is an omitted proof of the exact statement that makes the invariant nontrivial. A reviewer cannot fully certify the central theorem without access to or verification of [9]. The reader's CONDITIONAL verdict is therefore appropriate; my stress test does not change it.","tokens_in":12699,"tokens_out":27378,"duration_ms":233593,"concrete_test":"Independently verify Proposition 2 computationally and analytically. For m=3,4, enumerate all braids b of length ≤6 and all conjugating words a of length ≤4; compute C(b^r) and C((a^{-1} b a)^r) directly from the crossing-matrix definition, and test whether they differ by a simultaneous row/column permutation. Any failure falsifies Theorem 1. For a proof-level check, attempt to show C(a^{-1} b a)=P^{-1}C(b)P for pure b using the closure/linking interpretation of C; if no such derivation can be reconstructed without extra assumptions, the theorem should remain conditional on [9].","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's main invariant P(b) and E(b) are Hurwitz invariants only because of Proposition 3 (Section 3), quoted from [9] without proof: for a braid b whose permutation has order r, the rank, determinant, characteristic polynomial, and eigenvalues of C(b^r) are invariant under conjugation. This is the sole engine linking crossing-matrix spectral data to Hurwitz equivalence; Corollary 1 and Theorem 1 are immediate formal consequences of it. No derivation, reference to a published proof, or independent verification is supplied here. If Proposition 2/3 were false — e.g., if conjugating by a non-pure braid changed C(b^r) by more than a simultaneous row/column permutation — then P(b), Corollary 2, and Theorem 2 would all fail. The surface-link interpretation additionally imports Kamada's four-dimensional Markov theorem (Theorem 3, from [6]) without checking it, but the algebraic invariance itself depends only on Proposition 3. The examples are numerically consistent (I re-derived Example 10's polynomial (x+1)^3(x-3) by hand), so the internal computations are not the weak point; the unproved external premise is.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the crossing matrix C(b) of a braid b and, for a braid system \\vec b=(b_1,\\dots,b_n)\\in (B_m)^n, defines P(\\vec b) as the product of the characteristic polynomials of C(b_i^{r_i}), where r_i is the order of the braid permutation of b_i. The main theorems assert that P is invariant under Hurwitz equivalence (Theorem 1), that P factors into mn real linear factors whose roots sum to zero (Corollary 2), and that the essential eigenvalue multiset E(\\vec b), obtained by deleting eigenvalues 0, \\pm 1, is invariant under Hurwitz action, global conjugation, and stabilization/destabilization (Theorem 2). The paper presents the invariant as an obstruction to Hurwitz equivalence and, via Kamada's four-dimensional Markov theorem, as an indicator for the necessity of Euler fusion or fission. Two worked examples (Examples 9 and 10) illustrate that the invariant is easily computable and can be more discriminating than the trace product and monodromy group.","tokens_in":12887,"tokens_out":10075,"duration_ms":101246,"significance":"The central observation — that the Hurwitz action preserves conjugacy classes, so any conjugacy invariant of braids gives a Hurwitz invariant by multiplying over entries — is elementary, but the paper packages it cleanly around crossing-matrix spectral data. The invariant is computable and the examples are convincing; Example 9 is a genuine demonstration that P is strictly finer than two classical necessary conditions. The surface-link application is potentially useful. The main weakness is the unproved external premise Proposition 2/3; if that is supplied, the paper is a solid, modest contribution.","major_comments":[{"comment":"The Hurwitz invariance of P and E rests entirely on Proposition 2/3: if b is conjugate to b' and r is the order of the braid permutation, then C(b^r) is permutation equivalent to C((b')^r). This is quoted from the authors' own preprint [9] without proof. The rest of the paper — Corollary 1, Theorem 1, Corollary 2, Lemma 2, Theorem 2 — is a formal consequence of this statement, so a failure of Proposition 2/3 would invalidate the main claims. Because [9] is not a published reference, I ask the authors to include a self-contained proof of Proposition 2 (and hence Proposition 3), or at least to reproduce the argument from [9] in an appendix, so that the central invariant does not depend on an unchecked claim.","section":"§3, Proposition 2/3 (and §6, Theorems 1 and 2)"}],"minor_comments":[{"comment":"The displayed formula P(σ_m)=P(σ_m^{-1})=x^{m-2}(x+1)(x-1) is inconsistent with Example 6 when σ_m is viewed in B_{m+1}; it should be x^{m-1}(x+1)(x-1). The conclusion about E is unaffected, since the extra roots are 0 and ±1.","section":"Lemma 2 (III)"},{"comment":"In the definition of Euler fission, the final term is written τ(b'_{l+p}); this should presumably be τ(b'_{l+q}). The transformations (IV)/(IV') would also benefit from a precise statement of how q is related to the τ values.","section":"Theorem 3 (IV)"},{"comment":"Footnote 4 contains a typo ('compareing'). Also, reference [7] is an arXiv preprint; if a published or updated version exists, it should be cited.","section":"Reference list / footnote 4"},{"comment":"It would help to indicate the braid permutation orders explicitly for the two braids before displaying the fourth-power crossing matrices, since Proposition 2 uses r.","section":"Example 2"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a well-written, modest contribution. The only substantive issue is the missing proof of Proposition 2. If the authors provide a proof or a published reference, I would recommend acceptance. I do not see an internal inconsistency in the algebraic part; Example 10's computations check out."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. The paper does what it says: the product of characteristic polynomials of crossing matrices C(b_i^{r_i}) is invariant under Hurwitz equivalence, and the essential eigenvalue set is invariant under Hurwitz action, global conjugation, and stabilization. The proofs are short and correct given the premises. The novelty is modest — the construction is a direct corollary of the fact that the Hurwitz action preserves conjugacy classes of entries, so any conjugacy invariant gives a Hurwitz invariant by taking products or multisets — but the specific P(b) and E(b) are new and the one-way Euler-fusion indicator is a nice idea.\n\nThe soft spot is the load-bearing premise. Proposition 3, which says the rank, determinant, characteristic polynomial, and eigenvalues of C(b^r) are invariant under conjugation, is quoted from the authors' own preprint [9] and not proved here. If that statement failed, Theorem 1, Corollary 2, and Theorem 2 would all collapse. That is a genuine gap in the paper as written. It may well be true — the pure-braid case is in the literature, and the general case is plausible — but a referee should be able to check it without chasing an arXiv preprint. The examples in Sections 6 and 7 also contain polynomial computations with no derivations; I re-derived one of them by hand and it checked out, so I don't suspect errors, but as written the 'effective invariant' claim is asserted rather than demonstrated. Minor issue: the surface-link application imports Kamada's Markov theorem without comment, which is fine but worth a caveat.\n\nWho it's for: low-dimensional topologists working with surface braids and surface links, especially anyone who wants a computable necessary condition for avoiding Euler fusion/fission. It's a useful addition to the toolkit, not a breakthrough. I'd send it to a serious referee, but with a request that the authors either prove Proposition 3 or at least state it as a lemma and give a self-contained argument. The paper is honest, clearly written, and the internal logic holds.","headline":"A clean, short construction whose load-bearing conjugacy lemma is unproved here and outsourced to the authors' own preprint; worth refereeing if that lemma is supplied.","tokens_in":13431,"tokens_out":2065,"would_cite":false,"duration_ms":20909,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20F36","57K10"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that the product of characteristic polynomials of crossing matrices C(b_i^{r_i}) is invariant under Hurwitz equivalence, and that the resulting essential eigenvalue set detects when Euler fusion or fission is unavoidable be","keywords":["braid systems","Hurwitz equivalence","crossing matrix","characteristic polynomial","surface links","surface braids","Euler fusion","conjugacy invariant"],"falsifier":"The most direct check: take any braid b and a conjugate b' = a^{-1} b a; compute the characteristic polynomials of C(b^r) and C((b')^r). The paper predicts they coincide; a mismatch would refute the quoted Proposition 3 and thereby Theorem 1. Since the crossing matrices are finite integer matrices, this is a direct finite computation.","tokens_in":12515,"feed_emoji":"🪢","tokens_out":6990,"duration_ms":68987,"temperature":0.7,"pith_summary":"Braid systems—ordered collections of braids—are the combinatorial data underlying surface braids and surface links, and two systems are considered equivalent when they are related by the Hurwitz action. This paper proves that a single polynomial P(b), the product of the characteristic polynomials of the crossing matrices C(b_i^{r_i}) for each component, is invariant under the Hurwitz action. The key structural reason is that each b_i^{r_i} is a pure braid, so its crossing matrix is symmetric, has real eigenvalues, and its characteristic polynomial is unchanged by conjugation. Because P factors into real linear terms, the roots can be read off and manipulated easily; in the paper's worked example P distinguishes two four-component systems that the usual trace-product, monodromy-group, and permutation invariants cannot separate. The same construction produces an essential eigenvalue set that is unchanged by Hurwitz moves, global conjugation, and stabilization, so for surface links any difference in that set forces an Euler fusion or fission somewhere in the equivalence sequence.","feed_headline":"Polynomial invariant tells Hurwitz-inequivalent braids apart","feed_subtitle":"A product of crossing-matrix characteristic polynomials gives a computable test for surface-braid equivalence.","key_machinery":"The central object is the crossing matrix C(B) of a braid diagram, whose (i,j)-entry is the number of positive crossings minus the number of negative crossings in which strand i passes over strand j. It is well-defined on the braid, and for a pure braid it is symmetric. For a braid b whose permutation has order r, b^r is pure, so the symmetric integer matrix C(b^r) has real eigenvalues; its characteristic polynomial P(b) is a conjugation invariant by the quoted Proposition 3. The machinery of the paper is to multiply these single-braid polynomials across a braid system to obtain P(b), and to delete the trivial eigenvalues 0, ±1 to get the essential set E(b), which is stable under stabilizati","core_discovery":"The central claim is Theorem 1: if two braid systems are Hurwitz equivalent, their P-polynomials are equal. The construction: for a braid b_i with permutation order r_i, form the pure braid b_i^{r_i}, take its crossing matrix, and record the characteristic polynomial det(xI - C(b_i^{r_i})); multiply these n polynomials. Since the single-braid characteristic polynomial is a conjugacy invariant, each Hurwitz move—which replaces one component by a conjugate and another by a conjugate product—leaves the product untouched. Theorem 2 then strips off the eigenvalues 0 and ±1, which are the only eigenvalues that stabilization adds or removes, and shows the remaining multiset E(b) is invariant under","pith_inferences":["Editorial extension: because P is a product, it records only the union of the single-component spectra; the authors do not exploit correlations between components. A joint polynomial built from all C(b_i^{r_i}) simultaneously might separate additional Hurwitz classes.","Editorial extension: the structure used is just 'symmetric integer matrix attached to each group element, with a conjugacy-invariant characteristic polynomial,' so the same construction could produce Hurwitz invariants for other groups equipped with such a representation.","Editorial extension: in the paper's own example, P separates systems that agree on trace product, monodromy group, and all permutation/homomorphism projections; a natural next test is to survey random pairs of braid systems and measure how often P detects non-equivalence relative to those classical invariants."],"forward_implications":["P(b) and E(b) can be computed directly from braid words by forming crossing matrices, so the invariants are practical for small systems.","If P differs, Hurwitz equivalence is impossible; the paper demonstrates this on a pair that classical invariants fail to separate.","P(b) always factors into mn real linear factors whose roots sum to zero, giving a multiset of real numbers attached to the braid system.","For equivalent surface links, a difference in E(b) forces at least one Euler fusion or fission in any sequence of moves; E can therefore certify that the four-dimensional move is necessary."],"fun_headline_variants":["Crossing-matrix polynomial: Hurwitz-invariant for braid systems","Hurwitz moves leave this polynomial untouched, so braids differ","New polynomial distinguishes braid systems under Hurwitz moves","Invariant polynomial reveals Hurwitz-inequivalent braid systems","Braid systems separated by Hurwitz-invariant polynomial"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The main theorem inherits its force from the quoted Proposition 3, which says the characteristic polynomial of C(b^r) is invariant under conjugation; the present paper does not prove this, and an error there would collapse Theorem 1 and the surface-link application.","fun_headline_variants_meta":{"raw":{"variants":["Crossing-matrix polynomial: Hurwitz-invariant for braid systems","Hurwitz moves leave this polynomial untouched, so braids differ","New polynomial distinguishes braid systems under Hurwitz moves","Invariant polynomial reveals Hurwitz-inequivalent braid systems","Braid systems separated by Hurwitz-invariant polynomial"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000922,"raw_usage":{"total_tokens":3709,"prompt_tokens":578,"completion_tokens":3131,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":322,"completion_tokens_details":{"reasoning_tokens":3046}},"tokens_in":322,"tokens_out":3131,"duration_ms":22455,"temperature":1.0,"reasoning_tokens":3046,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T12:32:49.065649+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"The most direct check: take any braid b and a conjugate b' = a^{-1} b a; compute the characteristic polynomials of C(b^r) and C((b')^r). The paper predicts they coincide; a mismatch would refute the quoted Proposition 3 and thereby Theorem 1. Since the crossing matrices are finite integer matrices, this is a direct finite computation.","supporting_citations":[],"review_version":1}