{"id":"082c514f-2dea-49ed-9d93-ed511a30ea1b","arxiv_id":"2411.15614","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Using topological skew braces, the paper constructs concrete non-discrete topological biquandle structures on R^3 and S^1 x R^2 that are not quandles, and it computes the trefoil coloring space for one of the structures.","lead":"This short note uses skew braces, algebraic structures carrying two compatible group laws, to build new examples of topological biquandles: spaces whose two continuous operations can color knots. It produces explicit structures on R^3 and on S^1 times R^2, including non-involutive ones, and computes the trefoil coloring space for one example.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The examples depend on Theorem 2.12, whose Yang-Baxter verification is outsourced to [11,20,5] and whose proof in this paper omits the S-map/diagonal check; if that cited theorem does not cover full biquandle axioms, Corollaries 3.2 and 3.7 are unsupported.","rationale":"I read the paper in good faith and traced the central claim through Theorem 2.12 and the explicit formulas in Corollaries 3.2 and 3.7. The displayed operations for R^3 are continuous, bijective in the required slices, and the S-map takes the stated diagonal form; the computations in Theorem 3.1 and the formulas for r1 and r2 check out. The same structure for S^1×R^2 is visibly continuous, and the formulas appear consistent with the semidirect-product group law. The most load-bearing unresolved point is not the topology but the algebra: the Yang-Baxter step of Theorem 2.12 is delegated to the literature, while the biquandle definition also requires the S-map and diagonal condition. If the cited theorem covers exactly this biquandle construction, the paper's central examples are correct; if it only establishes a non-degenerate set-theoretic solution, the examples lack a complete proof. The reader identified the same weakest assumption, and the proposed direct verification of the explicit formulas resolves the dependence on the citation. The false identification of (S^1×R^2,∘) with SL(2,R) and the overstated novelty claim are real but non-load-bearing: the examples use the displayed formulas, not the group name, and the abstract only claims construction, not priority. Therefore I do not change the reader's conditional verdict.","tokens_in":10201,"tokens_out":22271,"duration_ms":205776,"concrete_test":"Verify Theorem 2.12 directly on the two R^3 examples without invoking [11]: substitute the displayed r1 and r2 into the Yang-Baxter equation and reduce the resulting polynomial identities to 0; then define S by S(b⋆a,a)=(a∗b,b), solve for b in terms of a and the first argument, and check S(a,a)=(τ(a),τ(a)) with τ(a)=−a′. A symbolic computation using a CAS such as sympy settles the issue; if the identities hold for these examples, the central claim is secure, and only the citation gap and the SL(2,R)/SE(2) mislabel remain as editorial issues.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central construction is mediated entirely by Theorem 2.12: Corollaries 3.2 and 3.7 are obtained by substituting the two brace structures into the operations a∗b=(−a+a∘b)′∘a∘b and b⋆a=−a+a∘b. The paper's proof of Theorem 2.12 establishes bijectivity of the slice maps and states the diagonal form S(a,a)=(−a′,−a′), but the Yang-Baxter equation is simply referred to [11, Thm 3.1], [20, Thm 4.1] and [5, Thm 2]. What is needed is not an arbitrary non-degenerate solution but the full biquandle package: YBE, existence and bijectivity of S, and the diagonal condition. If the cited result concerns only set-theoretic solutions, or uses a different map r, the four explicit formulas are not justified. The topological part is the weaker risk: each r is a composition of continuous polynomial or trigonometric maps, so continuity and the required inverse slices are visible from the displayed formulas. Hence the load-bearing assumption is the algebraic theorem, and its support in the manuscript is a citation plus a sketch.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs new examples of topological biquandles using skew braces. After recalling the definition of a biquandle and a skew brace, it states a theorem (Theorem 2.12) asserting that every skew brace (A,+,∘) yields a biquandle with operations b ⋆ a = -a + a ∘ b and a ∗ b = (-a + a ∘ b)' ∘ a ∘ b. The paper then verifies that the Heisenberg group (R^3,+,∘) and its opposite brace (R^3,∘,+) are skew braces, and it derives two explicit biquandle structures r1 and r2 on R^3. It similarly verifies that (S^1×R^2,+,∘) and (S^1×R^2,∘,+) are skew braces, where the second group is a semidirect product, and obtains two biquandle structures on S^1×R^2. The paper also discusses a coloring invariant J_X(K) for knots and gives the explicit coloring space for the trefoil using the S^1×C example.","tokens_in":7,"tokens_out":26030,"duration_ms":277951,"significance":"If the constructions are correct, the paper provides explicit, non-discrete, non-quandle topological biquandle structures on R^3 and S^1×R^2, which is a useful contribution because few such explicit examples are in the literature. The brace verifications in Theorems 3.1 and 3.6 are direct and checkable, and the formulas for r1 and r2 are explicit enough to be used in further work. The paper also correctly emphasizes that topological biquandle colorings give topological-space-valued knot invariants via the fixed-point space J_X(K). However, the claim that no concrete nontrivial topological biquandles were previously known is not accurate, and one of the groups used is misidentified.","major_comments":[{"comment":"The statement 'at present we have no concrete nontrivial examples of topological biquandle' is incorrect. Example 2.4 in the paper itself defines Alexander biquandles on any Z[t^{±1},s^{±1}]-module. Taking X = R with the usual topology and letting t and s act as multiplication by 2 gives a biquandle with operations a ∗ b = 2a - 3b and b ⋆ a = 2b. These operations are continuous, the topology is non-discrete, and for s ≠ 1 the structure is not a quandle in the sense used in the paper (b ⋆ a is not the trivial projection). Thus concrete nontrivial topological biquandles already exist. The author should correct this claim or define 'nontrivial' more restrictively.","section":"Introduction, §1"},{"comment":"Theorem 2.12 is the engine of the paper, since Corollaries 3.2 and 3.7 are obtained by substituting the two brace structures into its formulas. The proof, however, is a sketch: the Yang-Baxter equation is delegated to [11, Theorem 3.1], [20, Theorem 4.1], and [5, Theorem 2], while the S-map and diagonal conditions are only stated, not fully derived. Because the examples collapse if the cited theorem uses a different map r or covers only set-theoretic solutions, the paper should either provide a complete proof of Theorem 2.12 (including the full YBE identity) or state the precise theorem from the references and explicitly verify that it yields the same operations b ⋆ a = -a + a ∘ b and a ∗ b = (-a + a ∘ b)' ∘ a ∘ b, together with the S-map condition and the diagonal formula S(a,a) = (-a',-a').","section":"Theorem 2.12"},{"comment":"The group (S^1×R^2,∘) with the operation (e^{iθ_1},x_1,y_1) ∘ (e^{iθ_2},x_2,y_2) = (e^{i(θ_1+θ_2)}, x_2 cos θ_1 - y_2 sin θ_1 + x_1, x_2 sin θ_1 + y_2 cos θ_1 + y_1) is the Euclidean group SE(2) = SO(2) ⋉ R^2, not SL(2,R). The inverse formula given in the paper is the inverse in SE(2). SL(2,R) is a simple Lie group and is not isomorphic to this solvable semidirect product. This misidentification should be corrected; it does not affect the skew brace verification, but it is a substantive mathematical error in the text.","section":"§3.2, Theorem 3.6"}],"minor_comments":[{"comment":"In the displayed computation of (a_1+a_2) ∘ a'_1 ∘ (a_1+a_3), the first coordinate line has a misplaced parenthesis: the expression should be written as (e^{iθ_2}, -x_1 cos θ_1 cos(θ_1+θ_2) - ... + x_1 + x_2, (-x_1 cos θ_1 - y_1 sin θ_1) sin(θ_1+θ_2) + ... + y_1 + y_2). As typeset, the closing parenthesis after 'x_1 + x_2,' is missing, making the computation hard to follow.","section":"§3.2, proof of Theorem 3.6"},{"comment":"The corollaries state the biquandle structures via the maps r1 and r2, but they do not explicitly display the binary operations ∗ and ⋆. Writing out (x_1,y_1,z_1) ∗ (x_2,y_2,z_2) and (x_2,y_2,z_2) ⋆ (x_1,y_1,z_1) would make the continuity of the operations and the non-quandle property immediately visible to the reader.","section":"Corollaries 3.2 and 3.7"},{"comment":"The definition of a topological biquandle requires only that the two operations be continuous, not that the map r be a homeomorphism. This is fine for the coloring invariant, but the paper could state explicitly that continuity of r follows from continuity of ∗ and ⋆, and that no additional topological transfer lemma is needed because the examples are given by explicit continuous formulas.","section":"Definition 2.7"},{"comment":"The phrase 'the biquandle is not a quandle' is ambiguous. A biquandle is a quandle in the sense of Example 2.4 when one of the operations is the trivial projection b ⋆ a = b. The paper should define 'nontrivial' precisely, especially in light of the Alexander biquandle examples, to avoid the false novelty claim.","section":"Introduction"}],"recommendation":"major_revision","confidential_remarks":"The explicit brace verifications and the resulting biquandle formulas appear correct, and the examples are genuinely useful. The main concerns are the overstated novelty claim (Alexander biquandles on R already give non-discrete, non-quandle topological biquandles) and the misidentification of (S^1×R^2,∘) as SL(2,R) rather than SE(2). In addition, the proof of Theorem 2.12 should be made self-contained or precisely matched to the cited results, since it carries the whole construction. These issues are fixable and do not invalidate the central computations, so I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my take on Cheng's note. The actual mathematics in Section 3 is correct, and the explicit biquandle structures are new: the formulas for r1 and r2 on R^3 from the Heisenberg group and on S^1 x R^2 from the Euclidean motion group, including the non-involutive ones, are not in the prior literature. I re-derived the brace identities in Theorems 3.1 and 3.6 and they check out; the corollaries follow from Theorem 2.12, whose contents are indeed proved in the cited Guarnieri--Vendramin, Smoktunowicz--Vendramin, and Chang--Nelson papers. The topological transfer is not stated as a general lemma, but continuity of the derived operations is immediate from continuity of the group operations and inversion. So the construction works.\n\nThe soft spots are real but not load-bearing. First, the introduction says no concrete nontrivial topological biquandles exist, but Example 2.4 of the same paper gives the Wada biquandle on R with the Euclidean topology, which is continuous and not a quandle. The Alexander biquandle on R^2 does the same. So the novelty framing is overstated; the new thing is the skew-brace method and the specific geometric examples, not the first topological biquandle. Second, the group (S^1 x R^2, o) is called SL(2,R), but the displayed law is the Euclidean group SE(2). They are diffeomorphic as manifolds, but they are not the same Lie group, and this error should be fixed. Third, the proof of Theorem 2.12 is a sketch with the Yang-Baxter part outsourced; that is acceptable in a short note because the cited sources do cover it, but the paper should state exactly which result it relies on.\n\nThe paper is a useful short note for anyone working on topological quandles/biquandles or on skew-brace constructions of Yang-Baxter solutions. It deserves a serious referee; the computations are correct and the examples are explicit and checkable. I would send it to review, with a request to correct the SE(2) label and tone down the novelty claim in the introduction.","headline":"Correct computations and genuinely new explicit biquandle structures, but the novelty claim is overstated and the group on S^1 x R^2 is misidentified as SL(2,R).","tokens_in":11052,"tokens_out":3198,"would_cite":true,"duration_ms":28139,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["57K12","16T25"],"pacs":[],"model":"deepseek-v4-flash","headline":"A skew brace is a set carrying two compatible group laws, and this paper shows that every skew brace yields a biquandle; applying that recipe to the Heisenberg group on $R^3$ and to the Lie group $S^1\\times R^2$ produces explicit…","keywords":["topological biquandle","skew brace","Yang-Baxter equation","knot invariant","Heisenberg group","coloring invariant","virtual knot","SL(2,R)"],"falsifier":"Verify the set-theoretic Yang-Baxter equation directly for the displayed $r_2$ formula on $R^3$ at a randomly chosen triple; the calculation is finite algebra and any mismatch refutes the biquandle claim. A topological mismatch would also show up as a change in $J_Y(K)$ when the same knot is presented by two diagrams related by a Reidemeister move.","tokens_in":9732,"feed_emoji":"🪢","tokens_out":10626,"duration_ms":92584,"temperature":0.7,"pith_summary":"The paper answers a question that had no concrete examples: are there topological biquandles that are neither discrete nor merely topological quandles with a trivial second operation? It claims yes, and builds them from skew braces, sets equipped with two compatible group operations. Applying the general brace-to-biquandle conversion to the Heisenberg group law on $R^3$ and to the $S^1\\times R^2$ Lie group gives explicit continuous biquandle operations; the same recipe also covers all odd-dimensional Euclidean spaces. Since a topological biquandle's coloring spaces $J_X(K)$ are knot invariants, the examples yield new topological-space-valued invariants of knots, with the trefoil's coloring space computed in closed form.","feed_headline":"Skew braces produce topological biquandles on R3 and S1 x R2","feed_subtitle":"The brace group laws become continuous knot-coloring operations, so knots get non-discrete, non-quandle invariants.","key_machinery":"The machinery is Theorem 2.12, the skew-brace-to-biquandle translation: for a skew brace $(A,+,\\circ)$, define $b\\star a=-a+a\\circ b$ and $a*b=(-a+a\\circ b)'\\circ a\\circ b$. The Yang-Baxter equation for the map $r(a,b)=(b\\star a,a*b)$ is inherited from known skew-brace theory, while the required bijectivity of the slice maps, the auxiliary map $S$, and the diagonal map $\\tau(a)=-a'$ are verified by explicit inverse formulas. In the examples, every formula is built from continuous group operations on a manifold, so the biquandle axioms hold and continuity is automatic, which is what upgrades the construction to a topological biquandle.","core_discovery":"The central claim is that the explicit formulas in Corollaries 3.2 and 3.7 define genuine topological biquandle structures. On $R^3$, take ordinary addition together with the Heisenberg product $(x_1,y_1,z_1)\\circ(x_2,y_2,z_2)=(x_1+x_2,y_1+y_2,z_1+z_2+x_1y_2)$; this is a skew brace in both orders, and Theorem 2.12 converts it into the biquandles with the displayed maps $r_1$ and $r_2$. Because all the operations involved are continuous in the Euclidean topology, the resulting biquandles are topological. The same argument on $S^1\\times R^2$, where $\\circ$ is the semidirect product realizing the group $SL(2,R)$, yields two more topological biquandles. These structures are nontrivial in the paper's sense: the topology is the usual non-discrete one, and none of the biquandles collapses to a quandle, since a skew-brace biquandle is a quandle only when the brace is trivial.","pith_inferences":["The paper checks continuity example-by-example rather than stating a general lemma; a natural general theorem would say that any skew brace whose two group laws are continuous group operations on a topological space yields a topological biquandle, and this would encompass both families presented here.","Any Lie group that admits a second continuous group law making the pair a skew brace becomes a source of topological biquandles, so the $R^{2n+1}$ and $S^1\\times R^2$ examples are likely the first members of a much larger family.","A direct next check would be to compute $J_X$ for the unknot and the Hopf link under the $r_2$ structures; the paper computes only the trefoil, and those two calibrating computations would show how discriminating the new invariants are."],"forward_implications":["On each of $R^3$ and $S^1\\times R^2$, there is one involutive and one non-involutive topological biquandle; because involutive solutions are invariant under crossing change, the non-involutive $r_2$ structures are the ones that can give nontrivial classical knot invariants.","The $R^3$ construction extends to $R^{2n+1}$ through the Heisenberg group $H_n$, so every odd-dimensional Euclidean space carries two topological biquandle structures.","For links, the coloring space $J_X(L)$ for $(R^3,r_2)$ is described by explicit equations involving crossing numbers $c_{ij}$, so the invariant records pairwise linking information.","For the trefoil with the $S^1\\times C$ version of $r_2$, the coloring space is explicitly $\\theta_1=\\theta_2$ together with $(1-e^{-i\\theta_1}+e^{-2i\\theta_1})(\\alpha_1-e^{-i\\theta_1}\\alpha_2)=0$, a concrete finite-dimensional space that can be compared with other knot invariants."],"supporting_citations":[{"why":"Supplies the theorem that every skew brace gives a non-degenerate set-theoretic solution of the Yang-Baxter equation, which carries the YBE part of Theorem 2.12.","marker":"[11]"},{"why":"Provides an independent source for the same skew-brace-to-solution theorem, cited as alternate support for Theorem 2.12.","marker":"[20]"},{"why":"Gives a third source for Theorem 2.12 and connects skew-brace solutions to virtual link invariants, motivating the coloring perspective.","marker":"[5]"},{"why":"Introduces topological quandles and the coloring-space invariant that Theorem 2.9 extends to the topological biquandle setting.","marker":"[18]"},{"why":"Establishes the biquandle axioms and the biquandle coloring invariant for virtual links used throughout the paper.","marker":"[10]"}],"fun_headline_variants":["Skew braces produce topological biquandles on R3 and S1 x R2","Topological biquandles from skew braces on R3 and S1 x R2","Braces yield continuous knot colorings on R3 and S1 x R2","Nontrivial biquandles from skew braces on R3 and S1 x R2","Skew brace laws give topological biquandles on R3 and S1 x R2"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that Theorem 2.12 applies without extra hypotheses and that the derived biquandle operations are automatically continuous whenever the two brace group operations are continuous.","fun_headline_variants_meta":{"raw":{"variants":["Skew braces produce topological biquandles on R3 and S1 x R2","Topological biquandles from skew braces on R3 and S1 x R2","Braces yield continuous knot colorings on R3 and S1 x R2","Nontrivial biquandles from skew braces on R3 and S1 x R2","Skew brace laws give topological biquandles on R3 and S1 x R2"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000951,"raw_usage":{"total_tokens":3982,"prompt_tokens":794,"completion_tokens":3188,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":410,"completion_tokens_details":{"reasoning_tokens":3071}},"tokens_in":410,"tokens_out":3188,"duration_ms":20271,"temperature":1.0,"reasoning_tokens":3071,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T14:10:03.716963+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Verify the set-theoretic Yang-Baxter equation directly for the displayed $r_2$ formula on $R^3$ at a randomly chosen triple; the calculation is finite algebra and any mismatch refutes the biquandle claim. A topological mismatch would also show up as a change in $J_Y(K)$ when the same knot is presented by two diagrams related by a Reidemeister move.","supporting_citations":[{"cited_title":"Guarnieri and L","cited_arxiv_id":null,"evidence_quote":"Supplies the theorem that every skew brace gives a non-degenerate set-theoretic solution of the Yang-Baxter equation, which carries the YBE part of Theorem 2.12."},{"cited_title":"Smoktunowicz and L","cited_arxiv_id":null,"evidence_quote":"Provides an independent source for the same skew-brace-to-solution theorem, cited as alternate support for Theorem 2.12."},{"cited_title":"Chang and S","cited_arxiv_id":null,"evidence_quote":"Gives a third source for Theorem 2.12 and connects skew-brace solutions to virtual link invariants, motivating the coloring perspective."},{"cited_title":"Rubinsztein, T opological quandles and invariants of links, J","cited_arxiv_id":null,"evidence_quote":"Introduces topological quandles and the coloring-space invariant that Theorem 2.9 extends to the topological biquandle setting."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the biquandle axioms and the biquandle coloring invariant for virtual links used throughout the paper."}],"review_version":1}