{"id":"3a9132de-f9ed-4de4-9025-93c2fa89fab3","arxiv_id":"2505.23507","paper_version":4,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Symmetric quandle associated groups are characterized: the underlying quandle's group is a central extension of the symmetric one with a free abelian kernel, and embeddability is equivalent.","lead":"This mathematics paper analyzes the groups associated to symmetric quandles, algebraic objects that encode unoriented knots. It shows how these groups relate to the groups of the underlying quandles and proves a complete group-theoretic characterization.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: Theorem 3.1 and the embeddability theorem are supported by standard, correct facts; the flagged Proposition 2.1 is not a real vulnerability.","rationale":"The reader correctly identified Proposition 2.1 as the step most worth checking, but it is not a genuine weakness. The proposition follows immediately from the presentation: in the abelianization, the only relations are [e_x] = [e_{x*y}], so the abelian group is freely generated by the orbit representatives. The proof of Theorem 3.1 uses this in a valid way: Lemma 3.7 establishes distinct orbits for the relevant elements, so the images are linearly independent, and the kernel is free abelian with the claimed basis. The embeddability theorem similarly relies only on standard facts plus Proposition 10.1, whose proof correctly uses the As(Q)-equivariance of ρ. I also checked a peripheral but nontrivial aspect of Theorem 5.2: the constructed Q avoids the identity element because a mod-2 exponent-sum homomorphism factors through a group with a twisted Wirtinger presentation, forcing every generator to have nonzero image. Thus the announced results stand. I therefore recommend no change to the reader's ACCEPT verdict, while noting that Proposition 2.1 could be proved explicitly for completeness.","tokens_in":19470,"tokens_out":22755,"duration_ms":221802,"concrete_test":"Re-derive Proposition 2.1 directly from the presentation of As(Q): abelianize the Wirtinger presentation and verify that the resulting abelian group has exactly the orbit representatives as a basis, with no further relations. Then apply this to a concrete multi-orbit example, e.g., the disjoint union of a trivial quandle and a connected quandle, and check that the abelianization is the free abelian group on the two orbit classes.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim, Theorem 3.1, holds under scrutiny. The only potentially load-bearing assumption flagged by the reader, Proposition 2.1, is both standard and correct: abelianizing the presentation of As(Q) gives exactly the free abelian group on the orbits of the right As(Q)-action, with no additional hidden relations. Consequently, the linear independence argument in Lemma 3.7 is sound, and the proof of Theorem 3.1 goes through. The kernel is central, generated by the stated elements, and free abelian on the claimed basis. Theorem 10.2 also follows: the comparison of coefficients in As(Q)Ab is valid because Proposition 10.1 gives a complete set of orbit representatives, and the equivariance of ρ under the As(Q)-action is correctly used. A separate possible concern about Theorem 5.2, that the constructed symmetric quandle might contain the identity element if a generator becomes trivial, is also resolved: the homomorphism F(X) → Z/2 sending each generator to 1 vanishes on every twisted Wirtinger relator, so it factors through G and shows each generator remains nonzero in G. No load-bearing flaw was found.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper investigates the associated group As(Q,ρ) of a symmetric quandle (Q,ρ) and its relation to the associated group As(Q) of the underlying quandle. Its main results are: (1) Theorem 3.1, which identifies the kernel Z(Q,ρ) of the canonical surjection π_Q: As(Q) → As(Q,ρ) as a central, free abelian subgroup with explicit basis {e_x e_{ρ(x)} | x ∈ C}, and shows Z(Q,ρ) ∩ [As(Q),As(Q)] = 1; (2) Theorem 4.2, which exhibits As(Q) as the pullback of As(Q,ρ) and As(Q)_Ab over As(Q,ρ)_Ab; (3) Corollary 5.3, a group-theoretic characterization of groups As(Q,ρ) as those admitting a twisted Wirtinger presentation; (4) Theorem 6.1, computing As(Q,ρ)_Ab ≅ (Z/2)^{⊕Λ1} ⊕ Z^{⊕Λ2} and identifying this with the first symmetric quandle homology; (5) Proposition 7.2, a left adjointness statement; (6) Corollary 8.2, expressing H_2(Q) for connected Q via stabilizers in As(Q,ρ); and (7) Theorem 10.2, showing that (Q,ρ) embeds in As(Q,ρ) iff Q embeds in As(Q). The paper is clearly structured and the arguments are mostly direct manipulations of the defining presentations.","tokens_in":19695,"tokens_out":34703,"duration_ms":328036,"significance":"If correct, this is a substantial structural contribution to the theory of symmetric quandles and their associated groups. The explicit central extension and pullback descriptions, the twisted Wirtinger characterization, and the abelianization computation are likely to be of use in computing quandle homology and in low-dimensional topological applications. The embeddability criterion is clean and answers a natural question. The paper also generalizes earlier results of Hasegawa for involutive quandles and connects to work of Majid–Rietsch on covering groups. The proofs are standard but careful, with explicit bases and verifiable statements; the main weakness is a number of small omitted justifications and typos that are local and easily repaired.","major_comments":[],"minor_comments":[{"comment":"Proposition 2.1 is stated without proof or reference, although it is used in a load-bearing way in Lemma 3.7 and Theorem 10.2. Please add a citation or a short proof: abelianizing the defining presentation of As(Q) yields the free abelian group on the orbits of the right As(Q)-action.","section":"§2, Proposition 2.1"},{"comment":"The proof asserts that the constructed symmetric quandle Q satisfies condition (1), Q ⊂ G \\ {1}, but does not justify it. This can be shown by the homomorphism F(X) → Z/2 sending each generator to 1, which factors through G because every twisted Wirtinger relator w^{-1}xw = y^ε is satisfied modulo 2; it follows that no generator maps to the identity in G.","section":"§5, proof of Theorem 5.2"},{"comment":"The derivation of the presentation of As(Q,ρ)_Ab is compressed, and it relies on Proposition 10.1 before that proposition is proved. Please spell out the elimination of the generators [s_{ρ(xλ)}] for λ ∈ Λ2 via [s_{ρ(xλ)}] = [s_{xλ}]^{-1} and note explicitly that Proposition 10.1 is independent and proved later.","section":"§6, proof of Theorem 6.1"},{"comment":"When defining η(g)(q) = g(s_q), the paper does not explicitly verify that η(g) is a morphism of symmetric quandles. This follows immediately from the defining relations s_{q*r} = s_r^{-1} s_q s_r and s_{ρ(q)} = s_q^{-1}, but the verification should be included.","section":"§7, proof of Proposition 7.2"},{"comment":"The sentence 'Suppose that [ex], [ey]' appears to be missing the inequality; it should read 'Suppose that [ex] ≠ [ey]'.","section":"§10, proof of Theorem 10.2"},{"comment":"In the first sentence of the proof, 'O(ρ(xλ)), O(ρ(xμ)) whenever λ , μ' should be 'O(ρ(xλ)) ≠ O(ρ(xμ)) for λ ≠ μ'.","section":"§3, Lemma 3.7"},{"comment":"In the statement of Proposition 7.1, the target should be (Conj(As(Q,ρ)), Inv(As(Q,ρ))) rather than (Conj(As(Q), ρ), Inv(As(Q,ρ))).","section":"§7, Proposition 7.1"}],"recommendation":"minor_revision","confidential_remarks":"The paper is mathematically sound and within the scope of the journal. I have no concerns about attribution or novelty; the authors appropriately cite prior work, including Hasegawa's thesis. The minor issues listed should be addressed in revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is a solid, workmanlike contribution to quandle theory. The main structural results on As(Q,ρ)—the central extension description (Thm 3.1), the pullback square (Thm 4.2), the abelianization computation (Thm 6.1), and the embeddability equivalence (Thm 10.2)—are new for symmetric quandles and are proved correctly. I checked the central extension and pullback arguments in detail; the kernel of As(Q)→As(Q,ρ) is indeed central and free abelian with the stated basis, and the commutator-subgroup isomorphism follows. The dependence on Proposition 2.1 (abelianization of As(Q) is free abelian on orbit representatives) is legitimate: that fact is standard and correct, even though the paper quotes it without proof.\n\nThe soft spots are concentrated in the presentation of two proofs. Theorem 5.2 omits the explicit verification that the quandle it constructs avoids the identity element, which is one of the conditions in the covering-group definition; a one-line homomorphism to Z/2 fixes this. Theorem 6.1 is terse: the claimed presentation of As(Q,ρ)Ab needs a fuller justification that no hidden relations appear, and it would be cleaner to defer to the orbit classification of Prop. 10.1 with a more explicit coefficient check. Corollary 5.3 is essentially a restatement of the defining presentation in disguise, but the packaging as 'twisted Wirtinger presentations' and the connection to Majid-Rietsch covering groups is useful. The citation practice is honest, including attribution to Hasegawa for the involutive case.\n\nBottom line: this is a competent paper that a quandle theorist should read. It deserves a serious referee and, after small revisions, publication. I would cite it if I worked on associated groups or symmetric quandles.","headline":"A sound and useful structure theory for associated groups of symmetric quandles; the central extension, pullback, abelianization, and embeddability theorems hold up under scrutiny.","tokens_in":20204,"tokens_out":4558,"would_cite":true,"duration_ms":48345,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20F05","20N02","08A05","19C09","57K12"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that for a symmetric quandle, the associated group of the underlying quandle is a central extension of the associated group of the symmetric quandle, with kernel free abelian on the products $e_x e_{\\rho(x)}$.","keywords":["symmetric quandle","associated group","central extension","twisted Wirtinger presentation","abelianization","quandle homology","embeddability","good involution"],"falsifier":"For a small finite symmetric quandle, compute the kernel of $\\pi_Q\\colon \\mathrm{As}(Q)\\to \\mathrm{As}(Q,\\rho)$ and check whether the displayed elements $e_x e_{\\rho(x)}$ with $x\\in C$ form a $\\mathbb{Z}$-basis: a nontrivial integer relation among them, or any torsion in the kernel, would disprove Theorem 3.1; equivalently, a nonzero element killed by the map $\\mathrm{As}(Q)\\to \\mathrm{As}(Q,\\rho)\\times \\mathrm{As}(Q)^{\\mathrm{ab}}$ would disprove the pullback theorem.","tokens_in":19304,"feed_emoji":"🪢","tokens_out":10355,"duration_ms":88756,"temperature":0.7,"pith_summary":"This paper studies the associated group $\\mathrm{As}(Q,\\rho)$ of a symmetric quandle — a quandle equipped with an involution compatible with the quandle structure — and shows that this group controls the associated group $\\mathrm{As}(Q)$ of the underlying quandle. The main structural result is that $\\mathrm{As}(Q)$ is a central extension of $\\mathrm{As}(Q,\\rho)$ whose kernel is a free abelian group spanned by the elements $e_x e_{\\rho(x)}$, one for each class of a natural equivalence relation on $Q$. From this the authors derive three characterizations of $\\mathrm{As}(Q)$ in terms of $\\mathrm{As}(Q,\\rho)$, including a pullback diagram involving the two abelianizations, and they compute the abelianization of $\\mathrm{As}(Q,\\rho)$ as a direct sum of copies of $\\mathbb{Z}$ and $\\mathbb{Z}/2\\mathbb{Z}$. They also prove that a group is $\\mathrm{As}(Q,\\rho)$ for some symmetric quandle exactly when it admits a twisted Wirtinger presentation, and that a symmetric quandle embeds into its associated group if and only if its underlying quandle does.","feed_headline":"Quandle groups are central extensions of symmetric ones","feed_subtitle":"A free abelian kernel and a pullback square let symmetric-quandle groups describe ordinary ones and compute homology.","key_machinery":"The central object is the associated group of a symmetric quandle, presented by generators $s_x$ and relations $s_y^{-1}s_xs_y=s_{x*y}$ and $s_{\\rho(x)}=s_x^{-1}$. The argument is carried by the kernel elements $e_x e_{\\rho(x)}$ inside $\\mathrm{As}(Q)$: Lemma 3.3 shows they are central, Lemma 3.4 shows they generate the kernel of $\\pi_Q$, and Lemma 3.7 uses the classical description of $\\mathrm{As}(Q)^{\\mathrm{ab}}$ to prove they are linearly independent. A secondary device is the equivalence relation on $Q$ generated by $x\\sim x*y$ and $x\\sim \\rho(x)$; its classes index the free abelian basis, and the same partition governs the $\\mathbb{Z}/2\\mathbb{Z}$ versus $\\mathbb{Z}$ summands in the abelianization of $\\mathrm{As}(Q,\\rho)$. The group-theoretic characterization is carried by twisted Wirtinger presentations, meaning presentations whose relations all have the form $w^{-1}xw=y^{\\varepsilon}$ with $\\varepsilon=\\pm1$.","core_discovery":"The central claim is that for every symmetric quandle $(Q,\\rho)$ the canonical surjection $\\pi_Q\\colon \\mathrm{As}(Q)\\to \\mathrm{As}(Q,\\rho)$ sending $e_x$ to $s_x$ is a central extension: its kernel $Z(Q,\\rho)$ lies in the center of $\\mathrm{As}(Q)$ and is freely generated by the products $e_x e_{\\rho(x)}$ as $x$ ranges over representatives of the equivalence classes generated by the $\\mathrm{As}(Q)$-action on $Q$ together with $\\rho$. Because this kernel meets the commutator subgroup trivially, $\\pi_Q$ induces an isomorphism of commutator subgroups, and the square formed by the two abelianization maps is a pullback diagram. From this the paper obtains three distinct descriptions of $\\mathrm{As}(Q)$ in terms of $\\mathrm{As}(Q,\\rho)$, a computation of $\\mathrm{As}(Q,\\rho)^{\\mathrm{ab}}$ as a sum of $\\mathbb{Z}/2\\mathbb{Z}$ and $\\mathbb{Z}$ factors, a twisted-Wirtinger-presentation characterization of groups arising as $\\mathrm{As}(Q,\\rho)$, and a proof that symmetric embeddability is equivalent to ordinary embeddability.","pith_inferences":["The twisted-Wirtinger characterization suggests that computational tools developed for Coxeter, Artin, and braid groups could be applied to $\\mathrm{As}(Q,\\rho)$ for the corresponding quandles; the paper opens this direction but does not develop it.","The embeddability criterion could serve as a quick obstruction in the study of set-theoretic Yang–Baxter solutions: for a symmetric quandle it suffices to test the underlying quandle's embeddability, since failure of either embeddability implies failure of the other.","By analogy with the quandle case, one might seek a Hopf-type formula for the second symmetric quandle homology $H_2(Q,\\rho)$ in terms of a presentation of $\\mathrm{As}(Q,\\rho)$; the paper computes only $H_1(Q,\\rho)$ this way."],"forward_implications":["For every symmetric quandle, $\\mathrm{As}(Q)$ sits in a central extension $0\\to \\mathbb{Z}^{\\oplus C}\\to \\mathrm{As}(Q)\\to \\mathrm{As}(Q,\\rho)\\to 1$, so $\\mathrm{As}(Q,\\rho)$ together with one 2-cocycle determines $\\mathrm{As}(Q)$.","The commutator subgroups are isomorphic, $[\\mathrm{As}(Q),\\mathrm{As}(Q)]\\cong [\\mathrm{As}(Q,\\rho),\\mathrm{As}(Q,\\rho)]$; in particular, when $\\mathrm{As}(Q)^{\\mathrm{ab}}\\cong \\mathbb{Z}$ there is a splitting $\\mathrm{As}(Q)\\cong [\\mathrm{As}(Q,\\rho),\\mathrm{As}(Q,\\rho)]\\rtimes \\mathbb{Z}$.","The abelianization of $\\mathrm{As}(Q,\\rho)$ is $(\\mathbb{Z}/2\\mathbb{Z})^{\\oplus \\Lambda_1}\\oplus \\mathbb{Z}^{\\oplus \\Lambda_2}$, and it is isomorphic to the first symmetric quandle homology $H_1(Q,\\rho)$.","For a connected quandle, the second quandle homology is expressible on the symmetric side: $H_2(Q)\\cong (\\mathrm{Stab}_{\\mathrm{As}(Q,\\rho)}(x_0)\\cap [\\mathrm{As}(Q,\\rho),\\mathrm{As}(Q,\\rho)])_{\\mathrm{ab}}$.","A symmetric quandle embeds into $\\mathrm{As}(Q,\\rho)$ exactly when its underlying quandle embeds into $\\mathrm{As}(Q)$."],"supporting_citations":[{"why":"Supplies the method by which associated groups of quandles are analyzed through quotients and pullbacks; the proof of Theorem 4.2 is described as similar to its arguments.","marker":"[3]"},{"why":"Eisermann's formula expressing $H_2(Q)$ via isotropy subgroups of $\\mathrm{As}(Q)$ is the basis for Corollary 8.2.","marker":"[14]"},{"why":"Hasegawa proved Theorem 3.1, Corollary 3.2, and Theorem 4.2 in the involutive case; this paper extends those statements to all symmetric quandles.","marker":"[17]"},{"why":"Kamada–Oshiro defined symmetric quandle homology and the universal property of $\\mathrm{As}(Q,\\rho)$, used for Proposition 6.2 and Proposition 7.2.","marker":"[29]"},{"why":"Kishimoto's pullback criterion for squares of groups is exactly what turns the commutator-subgroup isomorphism into Theorem 4.2.","marker":"[32]"},{"why":"Majid–Rietsch introduced covering groups and proved the central-extension statement for symmetric subquandles of conjugation groups, which Theorem 5.2 generalizes.","marker":"[40]"}],"fun_headline_variants":["Symmetric quandle groups are central extensions of underlying ones","Free abelian kernel found in symmetric quandle group extension","Symmetric quandle embeddability matches underlying quandle embeddability","Abelianization of symmetric quandle groups computed via kernel","Pullback square links symmetric and ordinary quandle groups"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The linear-independence arguments rely on the stated classical fact that $\\mathrm{As}(Q)^{\\mathrm{ab}}$ is freely generated by one generator per orbit of the $\\mathrm{As}(Q)$-action on $Q$; this fact is used without proof, and if it failed, the central-extension basis, the pullback theorem, and the embeddability theorem would not follow.","fun_headline_variants_meta":{"raw":{"variants":["Symmetric quandle groups are central extensions of underlying ones","Free abelian kernel found in symmetric quandle group extension","Symmetric quandle embeddability matches underlying quandle embeddability","Abelianization of symmetric quandle groups computed via kernel","Pullback square links symmetric and ordinary quandle groups"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000258,"raw_usage":{"total_tokens":1538,"prompt_tokens":860,"completion_tokens":678,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":476,"completion_tokens_details":{"reasoning_tokens":595}},"tokens_in":476,"tokens_out":678,"duration_ms":6313,"temperature":1.0,"reasoning_tokens":595,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T12:46:00.100194+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a small finite symmetric quandle, compute the kernel of $\\pi_Q\\colon \\mathrm{As}(Q)\\to \\mathrm{As}(Q,\\rho)$ and check whether the displayed elements $e_x e_{\\rho(x)}$ with $x\\in C$ form a $\\mathbb{Z}$-basis: a nontrivial integer relation among them, or any torsion in the kernel, would disprove Theorem 3.1; equivalently, a nonzero element killed by the map $\\mathrm{As}(Q)\\to \\mathrm{As}(Q,\\rho)\\times \\mathrm{As}(Q)^{\\mathrm{ab}}$ would disprove the pullback theorem.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the method by which associated groups of quandles are analyzed through quotients and pullbacks; the proof of Theorem 4.2 is described as similar to its arguments."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Eisermann's formula expressing $H_2(Q)$ via isotropy subgroups of $\\mathrm{As}(Q)$ is the basis for Corollary 8.2."},{"cited_title":"↑2, 10, 18","cited_arxiv_id":null,"evidence_quote":"Hasegawa proved Theorem 3.1, Corollary 3.2, and Theorem 4.2 in the involutive case; this paper extends those statements to all symmetric quandles."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Kamada–Oshiro defined symmetric quandle homology and the universal property of $\\mathrm{As}(Q,\\rho)$, used for Proposition 6.2 and Proposition 7.2."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Kishimoto's pullback criterion for squares of groups is exactly what turns the commutator-subgroup isomorphism into Theorem 4.2."},{"cited_title":"Majid and K","cited_arxiv_id":null,"evidence_quote":"Majid–Rietsch introduced covering groups and proved the central-extension statement for symmetric subquandles of conjugation groups, which Theorem 5.2 generalizes."}],"review_version":1}