{"id":"8057185a-b883-41e5-8069-f29ca596ba2d","arxiv_id":"2412.20081","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A group is the knot group of a surface-link, orientable or not, exactly when it admits a (2m,n)-presentation with inverses satisfying the weak ∂-condition, and an analogous statement holds for knot symmetric quandles.","lead":"This paper gives a complete algebraic description of which groups can be the knot group of a surface-link in 4-space, including non-orientable surfaces. It also proves the same kind of characterization for the knot symmetric quandle and uses it to construct surface-links with prescribed dihedral quandle structure.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The if-direction needs K_{2m} to be exactly the set of adequate braids, not merely generated by them; the paper cites the weaker-sounding statement, leaving the plat-closure construction with a possible gap.","rationale":"The reader's weakest assumption identified the imported plat-closure calculus as the load-bearing structure, which is close to my concern but not identical. My stress-test isolates a more specific juncture within that calculus: the identification of K_{2m} with the set of adequate braids, which the manuscript only states via a 'generated by' formulation. This is exactly the point that makes the sufficiency halves of Theorems 1.3 and 1.4 operational, and it is not proved inside the paper. The concern is concrete and checkable: if Brendle–Hatcher's theorem indeed equates K_{2m} with the set of braids obtained from loops in the wicket configuration space, then the if-direction goes through, and the omitted details in Theorem 1.4 are routine. If not, the central characterization has a real gap. Because the likely outcome is that the standard Hilden-subgroup identification is correct, I do not think the reader's conditional verdict should be tightened to a rejection; the paper should add the precise statement and either a proof or a more explicit citation. The numerical mismatch in Theorem 1.5 Case 2 remains a concern for the applications, but it is separate from the central characterization and does not change the verdict on Theorems 1.3 and 1.4.","tokens_in":16171,"tokens_out":19681,"duration_ms":208863,"concrete_test":"Independently verify the Brendle–Hatcher result as used in §2.3: compute the image of the map π_1(W_m,w_0) → B_{2m} sending a loop f to β_f, and compare it with the subgroup K_{2m} generated by the listed elements σ_1, σ_2σ_1σ_3σ_2, and σ_{2k}σ_{2k-1}σ_{2k+1}^{-1}σ_{2k}^{-1}. If the image is strictly larger than K_{2m}, or if some element of K_{2m} is not in the image, then the if-direction of Theorem 1.3 needs a new argument. If the sets coincide, the central sufficiency step is sound and the concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The sufficiency halves of Theorems 1.3 and 1.4 both rely on the following step: from a presentation whose weak ∂-condition gives ∏ b_i^{-1}σ_1^{ε_i}b_i ∈ K_{2m}, the proof chooses a braided surface S with braid system β_i = b_i^{-1}σ_1^{ε_i}b_i and forms its plat closure eS. This is valid only if βS = ∏ β_i is an adequate 2m-braid, so that the wicket surface A_S exists. The manuscript bridges this point by citing Brendle–Hatcher: 'K_{2m} is precisely the subgroup generated by adequate 2m-braids.' If that statement only means that the set of adequate braids generates K_{2m} as a subgroup, then membership in K_{2m} does not by itself make βS adequate, and the plat-closure construction in the if-direction can fail. The proof would need the stronger identification of K_{2m} with the set {β_f : f ∈ π_1(W_m,w_0)}. The same issue is compounded in the if-part of Theorem 1.4, which is presented only as an outline and does not verify in writing that the constructed braided surface yields a knot symmetric quandle isomorphic to the given (Q,ρ), nor that condition (3) forces the claimed component counts.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper gives algebraic characterizations of knot groups and knot symmetric quandles of surface-links, including non-orientable ones. The main results, Theorems 1.3 and 1.4, state that a group (or symmetric quandle) is realized by a (c,d)-component surface-link with Euler characteristic χ if and only if it admits a (2m,n)-presentation with inverses satisfying a weak ∂-condition, with χ = 2m−n, and with an abelianization (or component structure) of the form Z^c ⊕ (Z/2)^d. The proofs use plat closures of braided surfaces, the Hilden subgroup, and prior results of the author. As an application, the paper shows that every dihedral quandle with any good involution is realizable as the knot symmetric quandle of a surface-link, and it constructs an infinite family of 2-component P2-irreducible P2-links.","tokens_in":76,"tokens_out":8701,"duration_ms":142433,"significance":"If the main theorems are correct, they extend the classical González-Acuña–Kamada characterization of knot groups of orientable surface-links to all surface-links, and they provide the first algebraic characterization of knot symmetric quandles for non-orientable surface-links. The application to dihedral quandles is attractive and gives new realizations of symmetric quandles. The use of plat presentations, rather than closed 2-dimensional braids, is well suited to the non-orientable setting. The paper draws on substantial prior work of the author and others, and the central strategy is plausible; however, several load-bearing steps in the sufficiency directions need to be made precise before the results can be accepted.","major_comments":[{"comment":"The proof asserts that because ∏ b_i^{-1} σ_1^{ε_i} b_i ∈ K_{2m}, the braided surface S with braid system (b_i^{-1} σ_1^{ε_i} b_i) can be chosen to be adequate. This requires that every element of K_{2m} is an adequate 2m-braid. The cited Brendle–Hatcher statement, as quoted in Section 2.3, says only that K_{2m} is the subgroup generated by adequate 2m-braids; that does not, by itself, imply that a product of conjugates of σ_1 is adequate. The proof needs the stronger fact that the set of adequate 2m-braids is a subgroup of B_{2m} and equals K_{2m}, or an explicit construction of an adequate braided surface with the given boundary braid. This gap affects the if-part of Theorem 1.4 as well.","section":"Section 2.4, proof of Theorem 1.3 (if part)"},{"comment":"The proof of Theorem 1.4 is only an outline. In the if-part, the existence of an adequate braided surface S with X(eS) isomorphic to (Q,ρ) is asserted without addressing the adequacy issue noted above, and the step 'By the condition (3), eS is (c,d)-component' is not justified in the text; it should be argued explicitly, for example by invoking the structure of connected components of knot symmetric quandles established in the only-if part. Since Theorem 1.4 is one of the two main theorems, the proof should be written out in full rather than left as a rephrasing of Theorem 1.3.","section":"Section 3.4, proof of Theorem 1.4"},{"comment":"Propositions 4.4 and 4.5 give symmetric quandle presentations for dihedral quandles with the identity and half-antipodal good involutions, but their proofs are omitted with the note that they are similar to Proposition 4.3. These propositions are used in Cases 3–5 of the proof of Theorem 1.5, so they are load-bearing for the application. Since the proof of Proposition 4.3 is already quite involved, the reader cannot easily verify the omitted claims. Please include the proofs or provide a detailed derivation of the presentations.","section":"Section 4, Propositions 4.4 and 4.5"}],"minor_comments":[{"comment":"The phrase 'a configuration of m wicket' should read 'a configuration of m wickets', and several typos such as 'equivalene', 'similaly', and 'Hurewitz' appear throughout the paper.","section":"Section 2.3"},{"comment":"The variable g is used in the final sentence of the proof ('F is (c,d)-component and genus g') but is not defined. If g denotes the total genus in the sense that χ = 2(c+d) − g, this should be stated explicitly.","section":"Section 2.4, proof of Theorem 1.3 (if part)"},{"comment":"The inference that H_1(R^4\\F) ≅ Z^c ⊕ (Z/2)^d forces F to have exactly c orientable and d non-orientable components is left implicit. This follows from Alexander duality and the structure of H^2 of a closed surface, but the proof should cite it explicitly.","section":"Section 2.4, proof of Theorem 1.3 (if part)"},{"comment":"The section heading contains the typo 'Thoerem' instead of 'Theorem'. In Case 2, the notation 'x1 = x(x3 x2)^k_2' is garbled and should be rewritten with clear subscripts.","section":"Section 4.1"},{"comment":"The list of 16 relations contains an apparent typo/OCR artifact ('yy =← −y x−1'), and the subsequent simplification of the 14 relations to the dihedral quandle is very terse. Please clarify the notation and the steps of the computation.","section":"Section 4, proof of Proposition 4.3"}],"recommendation":"major_revision","confidential_remarks":"The paper relies heavily on the author's own prior results [22] and [23] for the plat presentation and the quandle presentation of plat closures. The critical adequacy step rests on the Brendle–Hatcher theorem; the author should either prove the needed subgroup property or state the precise form of the theorem. The proof of Theorem 1.4 is too sketchy for a main theorem of the paper. The paper is within the scope of math.GT and the main idea is promising, but the gaps should be repaired before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The short version: Yasuda extends Kamada's characterization of knot groups of orientable surface-links to all surface-links, and proves the analogous statement for knot symmetric quandles. The main theorems look right and are a natural completion of a program that was previously only partial. The paper deserves a serious referee, but there are a few gaps that need to be addressed before the application section is fully convincing.\n\nWhat's new: the weak ∂-condition using the Hilden subgroup, the non-orientable case for groups (Theorem 1.3), and the symmetric quandle version (Theorem 1.4). The only-if directions are standard but clean: plat closure, braid monodromy, Alexander duality, van Kampen. The paper is honest that Theorem 1.4 is a rephrasing of Theorem 1.3 and gives an outline rather than full detail, which is reasonable given the similarity.\n\nThe soft spots are real but not fatal. First, the if-part of Theorem 1.3 uses the fact that a product in the Hilden subgroup is adequate. The paper cites Brendle–Hatcher as saying K_{2m} is precisely the subgroup generated by adequate braids. If that only says the adequate braids generate K_{2m}, membership doesn't imply adequacy. The fix is to state explicitly that the set of adequate braids is a subgroup (it's the image of π_1 of the wicket space), so every element of K_{2m} is adequate. This is likely what Brendle–Hatcher prove, but the paper should say so.\n\nSecond, Propositions 4.4 and 4.5 are asserted without proof. They're used to prove Theorem 1.5, so the application rests on unproved claims. Maybe the proofs are truly similar to Prop 4.3, but a referee should ask for them or at least a reference.\n\nThird, the index in Theorem 1.5 Case 2 looks off. The presentation given with exponent k = n/2 seems to produce R_{2n} rather than R_n for n≡0 mod 4. The intended statement might still be true, but the construction needs checking.\n\nThe reliance on the author's prior plat presentation is fine; those are published results and not circular. The paper is a solid extension, not a breakthrough, but it fills a natural gap.\n\nFor whom: anyone working on surface-links, their groups, and quandle invariants. It's a good paper to have in the literature.\n\nRecommendation: send it out. It's not desk-reject material. The referee should focus on the if-part details and the dihedral quandle presentations.","headline":"A believable and natural extension of the surface-link group characterization to non-orientable surfaces and to symmetric quandles, but the application section has a few gaps that need fixing before the paper is fully trustworthy.","tokens_in":16974,"tokens_out":7998,"would_cite":true,"duration_ms":69997,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["57K45","57K10"],"pacs":[],"model":"deepseek-v4-flash","headline":"A group is the knot group of a surface-link exactly when it has a $(2m,n)$-presentation with inverses satisfying the weak $\\partial$-condition and the right abelianization; the same holds for knot symmetric quandles.","keywords":["surface-links","knot groups","symmetric quandles","plat closure","braided surfaces","Hilden subgroup","non-orientable surfaces","dihedral quandles"],"falsifier":"Take an explicit braided-surface presentation whose braid product lies in the Hilden subgroup, such as the $(4,2)$-presentation used in the paper to realize the dihedral quandle with the antipodal map, build its plat closure, and compute the knot symmetric quandle directly from the complement. The theorem predicts exactly the dihedral quandle with the antipodal involution on a surface-link with two non-orientable components and Euler characteristic 2; any mismatch in the quandle, component count, or Euler characteristic would falsify the characterization.","tokens_in":15957,"feed_emoji":"🪢","tokens_out":16561,"duration_ms":139167,"temperature":0.7,"pith_summary":"This paper proves a complete algebraic characterization of which groups and which symmetric quandles arise as invariants of surface-links, including non-orientable ones. A group $G$ is the knot group of a surface-link with $c$ orientable and $d$ non-orientable components and Euler characteristic $\\chi$ exactly when $G$ has a presentation of an even braided form---$2m$ generators, $n$ braid relations, and $m$ inverse-pairing relations---satisfying a weak product condition on the braids, with $\\chi=2m-n$ and abelianization $\\mathbb{Z}^c\\oplus(\\mathbb{Z}/2)^d$. The same statement, with quandle operations and a good involution replacing the group law, characterizes knot symmetric quandles; the involution pairs the orientable components and fixes the non-orientable ones. The proof works through plat closures of braided surfaces, so each algebraic presentation corresponds to an explicit geometric construction. This matters because it turns a geometric existence question into a checkable presentation condition, and it yields that every dihedral quandle with any good involution occurs as the knot symmetric quandle of some surface-link.","feed_headline":"Surface-link groups characterized by one presentation condition","feed_subtitle":"The criterion also covers symmetric quandles, which tell orientable from non-orientable components","key_machinery":"The load-bearing construction is the plat closure of an adequate braided surface. A braided surface of degree $2m$ has a braid system $(\\beta_1,\\dots,\\beta_n)$ with $\\beta_i=b_i^{-1}\\sigma_1^{\\varepsilon_i}b_i$; the plat closure caps it off with $m$ wickets, and adequacy means the boundary braid lies in the Hilden subgroup $K_{2m}$ of the braid group, the subgroup generated by adequate braids. The weak $\\partial$-condition is precisely the requirement that the product $\\prod_{i=1}^n b_i^{-1}\\sigma_1^{\\varepsilon_i}b_i$ belongs to $K_{2m}$, which makes the closure exist. For such a closure, the knot group and knot symmetric quandle have the $(2m,n)$-presentation with inverses, the Euler characteristic is $\\chi=2m-n$ because the $n$ branch points are subtracted from the $2m$-sheeted cover, and Alexander duality identifies the first homology of the complement with $\\mathbb{Z}^c\\oplus(\\mathbb{Z}/2)^d$.","core_discovery":"The central claim is Theorem 1.3: a group $G$ is the knot group of a $(c,d)$-component surface-link with Euler characteristic $\\chi$ if and only if, for some $m,n\\ge 0$, $G$ has a $(2m,n)$-presentation with inverses satisfying the weak $\\partial$-condition, $\\chi=2m-n$, and $G/[G,G]$ is isomorphic to $\\mathbb{Z}^c\\oplus(\\mathbb{Z}/2)^d$. The $c$ free factors of the abelianization record the orientable components, while the $d$ copies of $\\mathbb{Z}/2$ record the non-orientable ones. Theorem 1.4 is the symmetric-quandle analogue: $(Q,\\rho)$ is the knot symmetric quandle of such a surface-link exactly when it has such a presentation, $\\chi=2m-n$, and $Q$ has $2c+d$ connected components, with $\\rho$ swapping $c$ pairs of them and fixing the other $d$. The orientable-only case is recovered by strengthening the weak $\\partial$-condition to the $\\partial$-condition. As an application, every dihedral quandle $R_n$ with any of its good involutions---the identity, the antipodal map, or the half-antipodal maps---is realized as the knot symmetric quandle of some surface-link.","pith_inferences":["The finiteness of the presentation form suggests that surface-link groups of bounded complexity could be enumerated by searching braid tuples whose product lies in $K_{2m}$, giving data analogous to classical braid enumeration for links.","Because the weak $\\partial$-condition is a membership test in the Hilden subgroup, the characterization suggests viewing surface-link groups as exactly the groups carrying an adequate even presentation with prescribed abelianization, which may connect to algorithms for recognizing such groups among finitely presented ones.","A natural testable extension is to apply Theorem 1.4 to other families of symmetric quandles, not just dihedral ones, by writing down presentations whose braid product lies in $K_{2m}$; the half-antipodal realization shows the criterion is flexible enough to handle involutions with mixed fixed-point behavior.","The identity-involution criterion for reducible projective-plane knots could be turned into an obstruction: any surface-knot whose knot symmetric quandle has a non-identity good involution is irreducible, so computing symmetric quandles offers a route toward the classical conjecture on projective-plane knots."],"forward_implications":["Any group satisfying the three algebraic conditions is guaranteed to arise from an actual surface-link, so the theorem converts a geometric realizability question into a presentation check.","The symmetric-quandle version gives a complete realizability test for quandles with involutions, including the half-antipodal cases that force non-orientable components.","Because the abelianization fixes the component type, the first homology of the complement determines exactly how many components of a realized surface-link are orientable and how many are not.","The dihedral-quandle application shows that the full knot quandle alone does not separate all surface-links; the good involution carries the extra distinguishing information, as stated in Corollary 4.6.","The infinite family of 2-component $P^2$-irreducible $P^2$-links in Theorem 5.3 shows that the characterization is not empty and produces many non-orientable examples beyond the two previously tabulated ones."],"supporting_citations":[{"why":"Supplies the plat-closure theorem: every surface-link is ambiently isotopic to the plat closure of a braided surface, the starting point of both characterizations.","marker":"[22]"},{"why":"Provides the presentation formulas for the knot group and knot symmetric quandle of a plat closure, together with the symmetric-quandle Tietze moves used in the proof.","marker":"[23]"},{"why":"Gives the orientable-surface-link group characterization that this paper extends to non-orientable components.","marker":"[10]"},{"why":"Identifies the Hilden subgroup with the subgroup generated by adequate braids, which is what the weak $\\partial$-condition checks.","marker":"[3]"},{"why":"Establishes the correspondence between braided surfaces and braid systems up to slide equivalence, used to go from presentations back to surfaces.","marker":"[19]"},{"why":"Defines the braid group action on free groups that appears in the $(2m,n)$-presentations and underlies the classical braid-closure theorem for links.","marker":"[2]"},{"why":"Classifies the good involutions of dihedral quandles, which the final application uses to realize every dihedral symmetric quandle.","marker":"[15]"},{"why":"Shows the knot quandle of a 2-twist spin of a 2-bridge knot is a dihedral quandle, used in the construction of the antipodal case.","marker":"[7]"}],"fun_headline_variants":["Plat presentations classify all surface-link groups and quandles","All surface-link groups and quandles characterized by one presentation","Surface-link groups and quandles: a single presentation condition suffices","One presentation condition pins down surface-link groups and quandles","Plat presentation test: all surface-link groups and quandles, orientable or not"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The result depends on the construction that builds every surface-link by capping off a braided surface with wickets, and on the claim that this construction always yields the stated group and quandle presentations; if that construction missed any non-orientable surface, the characterization would not follow.","fun_headline_variants_meta":{"raw":{"variants":["Plat presentations classify all surface-link groups and quandles","All surface-link groups and quandles characterized by one presentation","Surface-link groups and quandles: a single presentation condition suffices","One presentation condition pins down surface-link groups and quandles","Plat presentation test: all surface-link groups and quandles, orientable or not"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001376,"raw_usage":{"total_tokens":5587,"prompt_tokens":971,"completion_tokens":4616,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":587,"completion_tokens_details":{"reasoning_tokens":4527}},"tokens_in":587,"tokens_out":4616,"duration_ms":30116,"temperature":1.0,"reasoning_tokens":4527,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T23:38:11.650569+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take an explicit braided-surface presentation whose braid product lies in the Hilden subgroup, such as the $(4,2)$-presentation used in the paper to realize the dihedral quandle with the antipodal map, build its plat closure, and compute the knot symmetric quandle directly from the complement. The theorem predicts exactly the dihedral quandle with the antipodal involution on a surface-link with two non-orientable components and Euler characteristic 2; any mismatch in the quandle, component count, or Euler characteristic would falsify the characterization.","supporting_citations":[{"cited_title":"A characterization of groups of closed orientable surfaces in 4-space","cited_arxiv_id":null,"evidence_quote":"Gives the orientable-surface-link group characterization that this paper extends to non-orientable components."},{"cited_title":"Configuration spaces of rings and wickets","cited_arxiv_id":null,"evidence_quote":"Identifies the Hilden subgroup with the subgroup generated by adequate braids, which is what the weak $\\partial$-condition checks."},{"cited_title":"Homology groups of symmetric quandles and cocycle invariants of links and surface-links.Trans","cited_arxiv_id":null,"evidence_quote":"Classifies the good involutions of dihedral quandles, which the final application uses to realize every dihedral symmetric quandle."},{"cited_title":"On the knot quandle of a fibered knot, finiteness and equivalence of knot quandles","cited_arxiv_id":null,"evidence_quote":"Shows the knot quandle of a 2-twist spin of a 2-bridge knot is a dihedral quandle, used in the construction of the antipodal case."}],"review_version":1}