{"id":"8ee09b36-158c-4bed-9347-b645720dc136","arxiv_id":"2608.01504","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"In every simply connected smooth 4-manifold and every genus at least 3, there are infinitely many topologically distinct smoothly embedded surfaces whose orientation-preserving ambient symmetries are trivial and whose Alexander module is arbitrary.","lead":"For every genus at least 3, every simply connected 4-manifold, and every classical knot, this paper builds infinitely many distinct genus-g surfaces with no ambient symmetries at all, while forcing their Alexander module to match the chosen knot. The construction uses repeated rim surgeries and shows that a certain nonabelian group invariant exactly records the surgery curves.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; the main dependency is the pair of cited knot-supply theorems in Proposition 15.5, both of which appear correctly applied.","rationale":"The reader's weakest_assumption identifies the external supply theorems as load-bearing; I agree that these are the least secure dependencies, since the universality over all knots and the infinite pairwise-distinct family both flow through Proposition 15.5. However, both cited theorems are established results, and the paper's application of them is mathematically sound: S-equivalence is correctly shown to preserve the full Alexander module, and increasing volumes give nonisomorphic meridian-marked knot groups by Mostow–Prasad. I found no internal error in the long centralizer/Bass–Serre argument, the filling-system construction, or the Alexander-module calculation. Thus the conditional verdict is appropriate: independent verification of the external supplies and the key computations is warranted, but there is no identified defect that would force rejection or revision of the central claim.","tokens_in":102,"tokens_out":55368,"duration_ms":1107840,"concrete_test":"Recompute the cokernel invariance in Proposition 15.5(i) on a concrete Seifert matrix, e.g., V=[1] for the trefoil, by applying the two displayed elementary-enlargement formulas and checking that the resulting Alexander module is isomorphic to the original up to the invertible 2×2 block; then confirm that the cited statements of [5, Theorem 1.1] and [10, Theorem 1.2] indeed provide a hyperbolic knot with an S-equivalent Seifert matrix and infinitely many zero-module hyperbolic knots with unbounded volume.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I read the full construction and found no internal inconsistency or circular step. The central claim rests, as the reader notes, on Proposition 15.5: Friedl's theorem ([5, Theorem 1.1]) must supply a hyperbolic knot whose Seifert matrix is S-equivalent to the given knot's, with S-equivalence preserving the full Λ-module cokernel of tV−V^T, and Kalfagianni's theorem ([10, Theorem 1.2]) must supply infinitely many Alexander-trivial hyperbolic knots with strictly increasing volumes. Both are published results, and the paper's reduction to the needed statements is correct: the row/column computations in §15.4 show that elementary enlargements change the Alexander presentation by an invertible direct summand, so the full module is preserved; Mostow–Prasad rigidity turns strictly increasing volumes into nonisomorphic meridian-marked groups. The dependency is real and is the weakest link in the chain, but it is a verification point rather than a demonstrated flaw. The internal arguments—ordered rim surgeries, centralizer classification, Bass–Serre recovery, the filling-system construction, and Alexander-module Mayer–Vietoris—are mutually consistent and appear to support the stated theorem.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves Theorem 1.1: for every g ≥ 3 and every classical knot K in S^3, there are infinitely many pairwise topologically inequivalent smoothly embedded genus-g surfaces in S^4 whose orientation-preserving extendable mapping class groups are trivial in both the topological and smooth categories and whose first Alexander module is isomorphic to A(K). The construction starts with the standard unknotted surface, performs preliminary rim surgeries along meridian-disk boundaries to make the push-off homomorphism injective, then performs an ordered sequence of rim surgeries along a filling chain of curves with consecutive geometric intersection one. The exterior groups are analyzed by Bass–Serre theory: each product vertex group is recognized intrinsically from the exterior group together with the positive meridian, and the centralizer classification in Sections 6–11 lets the reverse induction recover each surgery curve and descend to the previous exterior. A filling chain with trivial labelled stabilizer in Section 12 converts this into the statement that any extendable mapping class is the identity. The Alexander module is computed by Mayer–Vietoris in Section 15, and two published knot-supply theorems, Friedl's and Kalfagianni's, are used to prescribe the module and to produce infinitely many distinct surfaces. Corollary 1.2 transfers the construction to arbitrary closed simply connected smooth 4-manifolds.","tokens_in":63782,"tokens_out":15378,"duration_ms":145179,"significance":"If correct, this is the first construction of closed oriented positive-genus smoothly embedded surfaces in S^4 with trivial full orientation-preserving extendable subgroup, and the first with prescribed first Alexander module. The manuscript is exceptionally explicit about based push-off formulas, groupoid conventions, and the hypotheses needed at each induction step; the centralizer–transporter and cyclic-subgroup conjugacy conditions are stated as definitions, and the Bass–Serre arguments are self-contained. The main external input is Proposition 15.5, which uses Friedl's realization of Seifert matrices by hyperbolic knots and Kalfagianni's hyperbolic knots with trivial Alexander polynomial and arbitrarily large volume; both reductions appear correct, and I found no internal inconsistency or circularity in the recognition chain of Sections 6–11. The paper therefore makes a substantial contribution to the realization problem for extendable mapping class groups.","major_comments":[{"comment":"The construction of infinitely many pairwise inequivalent surfaces with prescribed Alexander module rests entirely on the two external supply theorems, [5, Theorem 1.1] and [10, Theorem 1.2]. I have checked the reductions: the elementary-enlargement computations do preserve the full Λ-module cokernel because the added 2×2 blocks have determinants t and 1, both units in Λ, and Mostow–Prasad rigidity converts strictly increasing volumes into pairwise nonisomorphic groups. The dependency is therefore real but not a demonstrated flaw. To make the paper easier to verify, please state the exact published formulations of the two theorems, including the precise class of Seifert matrices in [5] and the volume-growth condition in [10], and confirm that the meridian can be chosen compatibly with the paper's positive-meridian convention. This is a verification point rather than a correction to the internal argument.","section":"§15.4, Proposition 15.5"}],"minor_comments":[{"comment":"There are typographical spacing errors, most notably 'simply connected4-manifolds' in the title and several instances of 'S4' that should read 'S^4'; these should be fixed in the final version.","section":"Title and abstract"},{"comment":"In the row/column computation, please state explicitly that the two 2×2 blocks have determinants t and 1 respectively, both units in Λ, since the current wording asserts invertibility without giving the determinants and a reader might initially misread the first block as having determinant t outside the units of Z[t,t^{-1}].","section":"§15.4, Proposition 15.5"},{"comment":"The map κ_c is not injective, as the proof notes by allowing arc endpoints to move along the boundary; it would help to add one sentence after the statement that this non-injectivity is intentional and is controlled in Lemma 12.8 through the twist relationship.","section":"§12, Lemma 12.7"},{"comment":"After choosing K^♯ and the sequence L_r, it would be helpful to state explicitly that the fixed rank-two knots form a finite set and that the varying knot J_{i_0} is chosen from the L_r outside that finite set, so that the pairwise-nonisomorphism hypothesis of Theorem 14.1 is satisfied at every stage.","section":"§15.5, Proof of Theorem 1.1"}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is very long and intricate; I recommend that a second referee with expertise in both Bass–Serre theory and knot theory also see it. The only load-bearing point I would want explicitly verified by the authors is the exact statement of the two external supply theorems in Proposition 15.5; the internal reductions appear correct. I see no citation or novelty concern: the distinction from Baykur–Sunukjian's non-simply-connected ambient construction and from earlier results controlling only homology or finite subgroups is clearly drawn."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The headline is simple: this is the first construction I know of positive-genus smoothly embedded surfaces in S^4 whose full orientation-preserving extendable mapping class group is trivial, in both the smooth and topological categories, with the first Alexander module prescribed arbitrarily. That's a real advance on the small-groups side of the realization problem. The previous record from Liu-Ni-Sun-Wang and Q. Liu gave finite or homology-controlled extendable subgroups; Baykur-Sunukjian got trivial extendable groups but in a hyperbolic 4-manifold, not a simply connected one. If Theorem 1.1 is right, it settles the 'as small as possible' direction in every genus at least 3.\n\nWhat I find genuinely new is the method: an ordered sequence of rim surgeries where the traversal order is recorded in the push-off homomorphism, and a Bass-Serre centralizer-recognition theorem that recovers each surgery curve from the exterior group. It is a clever way to turn a nonabelian group invariant into a geometric rigidity statement. The paper is 84 pages of explicit lemmas and conventions, with no internal contradiction that I could find. The dependencies between sections are honest, and the final Alexander-module computation is direct. The reader's circularity worry does not land: the rigidity proof builds exterior groups, then proves recognition, then applies it.\n\nThe soft spots are proportionate. The main load-bearing point is Proposition 15.5, which relies on Friedl's theorem (any Alexander module from a hyperbolic knot) and Kalfagianni's theorem (Alexander-trivial hyperbolic knots with large volume). Both are published and the reduction to them looks correct, so this is a verification point, not a demonstrated flaw. The second flag is that the author says DeepSeek was used for algebraic computations in Propositions 8.4, 9.4, and 15.2. Those three propositions are not peripheral; they are central to the centralizer classification and the Alexander-module calculation. I would want independent, human-checkable proofs of those before accepting the whole chain. The paper is not machine-checked, so that verification burden is real.\n\nOverall, who is this for? Topologists working on surfaces in 4-manifolds, mapping class groups, and Alexander invariants. It deserves a serious referee. Send it to referees, with a request to focus on the AI-assisted computations and the quoted supply theorems. I would not desk reject this.","headline":"A long, serious construction that plausibly proves the first trivial extendable mapping class groups for surfaces in S^4 with prescribed Alexander module; worth a careful referee, with two verification flags.","tokens_in":64322,"tokens_out":2738,"would_cite":true,"duration_ms":24324,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["57K40","57K45","57R50"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every genus at least three and every knot K, infinitely many genus-g surfaces in S^4 have trivial extendable symmetry group and prescribed first Alexander module.","keywords":["knotted surfaces","rim surgery","extendable mapping class groups","Alexander modules","Bass–Serre theory","4-manifolds","mapping class groups","topological equivalence"],"falsifier":"The most direct check is to take the smallest case, g=3 with the unknot as the prescribed module, follow the explicit construction, and compute the meridian-fixing automorphisms of the final exterior group that intertwine the push-off homomorphism with a mapping class; the theorem predicts only the identity can occur, so exhibiting a nontrivial such pair would refute it. A second check is to test whether two different varying hyperbolic knots in the family have isomorphic meridian-marked groups; by rigidity of finite-volume hyperbolic 3-manifolds they would then have equal volume, so comparing the chosen strictly increasing volumes settles whether the distinguishing mechanism works.","tokens_in":63363,"feed_emoji":"🪢","tokens_out":9324,"duration_ms":82805,"temperature":0.7,"pith_summary":"The paper proves that ambient symmetry of an embedded surface can be switched off completely while an abelian knot invariant is prescribed. For every genus g≥3 and every classical knot K⊂$S^{3}$, it constructs infinitely many pairwise topologically inequivalent smoothly embedded genus-g surfaces in $S^{4}$ whose orientation-preserving extendable mapping class subgroups are trivial in both the smooth and topological categories, and whose first Alexander module is isomorphic to the Alexander module of K. The rigidity is detected entirely by nonabelian information in the exterior group; the Alexander data are prescribed independently. The construction uses an ordered sequence of ordinary untwisted rim surgeries with hyperbolic knots inside a 4-ball, so the same family embeds in every closed, connected, oriented, simply connected smooth 4-manifold.","feed_headline":"Rim surgery erases all ambient symmetry of knotted surfaces","feed_subtitle":"A construction in every genus ≥3 makes the full extendable symmetry group trivial while prescribing the Alexander module freely.","key_machinery":"The load-bearing mechanism is the one-step exterior-group amalgam produced by an ordinary untwisted rim surgery, together with the push-off homomorphism. If F' is obtained from F by rim surgery along d using a hyperbolic knot J, then G' = G *_{\\langle μ,h\\rangle}(K_J×\\langle h\\rangle), where μ is the positive meridian and h=ρ_F(d); there is a canonical retraction r:G'→G with r∘ρ_{F'}=ρ_F and kernel the normal closure of [K_J,K_J]. The new push-off inserts conjugates of the preferred longitude of J at each crossing, in traversal order, so the exterior group remembers the curve rather than only its homology class. Bass–Serre theory then yields the centralizer–transporter property for each edge subgroup, the cyclic-subgroup conjugacy condition used to recover curves, and a reverse induction descending from G_i to G_{i-1}; a filling curve system with trivial labelled stabilizer completes the rigidity.","core_discovery":"The central discovery is a full rigidity theorem. For every g≥3 and every classical knot K, there exist infinitely many pairwise topologically inequivalent smoothly embedded genus-g surfaces F_{g,n}⊂$S^{4}$ with E^+_TOP(F_{g,n})=E^+(F_{g,n})=1 and A_1(F_{g,n})≅A(K) as modules over Λ=Z[t,$t^{{-1}}$]. The method first makes the surface subgroup visible in the exterior: after preliminary rim surgeries the push-off homomorphism ρ_0:π_1(Σ_g)→π_1(E_{F_0}) is injective, and each further rim surgery along a curve d_i with hyperbolic knot J_i changes the exterior group by the amalgam G_i≅G_{i-1}*_{\\langle μ,h_i\\rangle}(K_{J_i}×\\langle h_i\\rangle). The nonabelian centralizer structure of these amalgams identifies the product vertex group B_i, hence the surgery curve d_i, from the exterior group together with the positive meridian; a labelled filling system with trivial stabilizer then forces any extendable mapping class to be the identity. The first Alexander module is a direct sum of the Alexander modules of the chosen knots, so it can be prescribed independently of the rigidity.","pith_inferences":["Inference: the same reverse-induction recognition should apply to any ordered sequence of rim surgeries whose curves form a labelled filling system with trivial stabilizer, so the method is not bound to the specific filling chain built here.","Inference: replacing the trivial labelled stabilizer by a prescribed finite subgroup of the mapping class group might yield surfaces whose extendable subgroup is exactly that subgroup, if a filling system with that stabilizer exists.","Inference: since the Alexander module is a direct sum of knot-module summands, one could prescribe a direct sum of several knots' Alexander modules, not just a single one, by using more prescribed-module knots at preliminary stages."],"forward_implications":["For every genus g≥3 there are infinitely many topologically distinct genus-g surfaces in S^4 with trivial full extendable mapping class group, including many with vanishing first Alexander module.","The same infinite families sit inside every closed, connected, oriented, simply connected smooth 4-manifold, because the entire construction is supported in a 4-ball.","Topological inequivalence of the surfaces implies smooth inequivalence and non-isotopy, so the family provides infinitely many distinct knotted embeddings.","The Alexander module can be prescribed arbitrarily while the rigidity remains intact; the distinguishing invariant is the nonabelian centralizer structure of the exterior group."],"supporting_citations":[{"why":"defines ordinary untwisted rim surgery, the local operation on which every surface in the construction is built.","marker":"[4]"},{"why":"supplies a hyperbolic knot realizing any prescribed knot Alexander module, needed to prescribe A(K).","marker":"[5]"},{"why":"supplies infinitely many hyperbolic knots with trivial Alexander module and pairwise non-isomorphic groups via volume, used to distinguish the surfaces.","marker":"[10]"},{"why":"malnormality of hyperbolic cusp subgroups underpins the centralizer and transporter calculations for knot group factors.","marker":"[6]"},{"why":"identifies topological and smooth mapping class groups of surfaces, used to define E^+_TOP and E^+.","marker":"[7]"},{"why":"uniqueness of topological normal bundles in dimension four normalizes the action of an exterior homeomorphism on push-off data.","marker":"[3]"},{"why":"provides the combinatorial group theory, amalgam normal forms, and Britton-lemma technology used throughout the Bass–Serre arguments.","marker":"[14]"},{"why":"is the direct precursor computing the extendable subgroup after a single rim surgery, which motivates the ordered-sequence construction.","marker":"[15]"}],"fun_headline_variants":["Rim surgery kills all ambient symmetries of genus-g surfaces","Trivial symmetry groups for infinite families of surfaces in 4-balls","Every knot's Alexander module realized by rigid embedded surfaces","Genus-fixed knotted surfaces with no ambient automorphisms","Rigid surfaces in S^4 with prescribed Alexander modules"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that two external supply theorems hold—every knot's Alexander data is realized by a hyperbolic knot, and infinitely many Alexander-trivial hyperbolic knots have pairwise non-isomorphic groups—because if either supply fails the infinite family with the prescribed Alexander module collapses.","fun_headline_variants_meta":{"raw":{"variants":["Rim surgery kills all ambient symmetries of genus-g surfaces","Trivial symmetry groups for infinite families of surfaces in 4-balls","Every knot's Alexander module realized by rigid embedded surfaces","Genus-fixed knotted surfaces with no ambient automorphisms","Rigid surfaces in S^4 with prescribed Alexander modules"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000387,"raw_usage":{"total_tokens":2317,"prompt_tokens":989,"completion_tokens":1328,"prompt_tokens_details":{"cached_tokens":896},"prompt_cache_hit_tokens":896,"prompt_cache_miss_tokens":93,"completion_tokens_details":{"reasoning_tokens":1242}},"tokens_in":93,"tokens_out":1328,"duration_ms":20334,"temperature":1.0,"reasoning_tokens":1242,"cache_read_input_tokens":896,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T15:07:08.941353+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"The most direct check is to take the smallest case, g=3 with the unknot as the prescribed module, follow the explicit construction, and compute the meridian-fixing automorphisms of the final exterior group that intertwine the push-off homomorphism with a mapping class; the theorem predicts only the identity can occur, so exhibiting a nontrivial such pair would refute it. A second check is to test whether two different varying hyperbolic knots in the family have isomorphic meridian-marked groups; by rigidity of finite-volume hyperbolic 3-manifolds they would then have equal volume, so comparing the chosen strictly increasing volumes settles whether the distinguishing mechanism works.","supporting_citations":[{"cited_title":"Fintushel and R","cited_arxiv_id":null,"evidence_quote":"defines ordinary untwisted rim surgery, the local operation on which every surface in the construction is built."},{"cited_title":"Friedl,Realizations of Seifert matrices by hyperbolic knots, J","cited_arxiv_id":null,"evidence_quote":"supplies a hyperbolic knot realizing any prescribed knot Alexander module, needed to prescribe A(K)."},{"cited_title":"Kalfagianni,Alexander polynomial, finite type invariants and volume of hyperbolic knots, Algebr","cited_arxiv_id":null,"evidence_quote":"supplies infinitely many hyperbolic knots with trivial Alexander module and pairwise non-isomorphic groups via volume, used to distinguish the surfaces."},{"cited_title":"de la Harpe and C","cited_arxiv_id":null,"evidence_quote":"malnormality of hyperbolic cusp subgroups underpins the centralizer and transporter calculations for knot group factors."},{"cited_title":"Hatcher,The Kirby torus trick for surfaces, Enseign","cited_arxiv_id":null,"evidence_quote":"identifies topological and smooth mapping class groups of surfaces, used to define E^+_TOP and E^+."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"uniqueness of topological normal bundles in dimension four normalizes the action of an exterior homeomorphism on push-off data."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the combinatorial group theory, amalgam normal forms, and Britton-lemma technology used throughout the Bass–Serre arguments."},{"cited_title":"Extendable mapping classes of knotted surfaces obtained by rim surgery in $S^4$","cited_arxiv_id":"2605.31383","evidence_quote":"is the direct precursor computing the extendable subgroup after a single rim surgery, which motivates the ordered-sequence construction."}],"review_version":2}