{"id":"343cec60-749a-4577-9b90-0da949f37242","arxiv_id":"2507.20360","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":1,"one_line_summary":"A state-sum invariant for immersed surface-links is defined using singular biquandle 3-cocycles and proven invariant under all generating moves including the singular move (h).","lead":"This paper introduces a new algebraic invariant that can distinguish some immersed surfaces in four-dimensional space from each other. If correct, it gives knot theorists a new tool for studying surfaces that pass through themselves.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Example 4.6 is invalid: the stated θ_S violates Definition 4.2(iii) for the singular-compatible pair (b,c)=(4,7) with a=0, so the claimed nontrivial value 72+9t is unsupported.","rationale":"The paper's central claim comprises the state-sum invariance theorem and the nontriviality demonstration. Theorem 4.5 is plausible but its proof is delegated to unreproduced figures; the example was intended to supply the missing evidence. The algebraic failure of θ_S is the most load-bearing concern because it is checkable from the text and directly falsifies the only nontriviality claim. The reader's conditional verdict remains appropriate: the paper should be revised to supply a correct singular cocycle and a written verification of the move-(h) invariance, including all cases. I do not claim the main theorem is false; rather, the paper as written does not establish a working example. The concrete test above settles the cocycle point.","tokens_in":8187,"tokens_out":26938,"duration_ms":267759,"concrete_test":"Use the quandle table in Example 4.6 to compute 4∗7 and 7∗4, then evaluate θ_S(0,4,7) and θ_S(0,7,4) from the stated support. Since 4∗7=4 and 7∗4=7, Definition 4.2(iii) applies to (b,c)=(4,7); the check is whether θ_S(0,4,7)=t equals θ_S(0,7,4)^{-1}=1. If the equality fails (as it does with the support as printed), either the support of θ_S contains an error or Definition 4.2(iii) is not the correct antisymmetry condition; in either case Example 4.6's 72+9t must be recomputed. This single computation settles whether the nontriviality claim lands.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Example 4.6 is the only demonstration that the new invariant is non-trivial, and its input θ_S is asserted to be a singular biquandle 3-cocycle. That assertion is false under the manuscript's own Definition 4.2(iii). In the quandle case used there, ▷=∗ and ▷ is first-factor projection, so the singular-compatibility hypothesis b▷c=b▷c, c▷b=c▷b reduces to b∗c=b and c∗b=c. In the table for Q=Z9, 4∗7=4 and 7∗4=7, so (b,c)=(4,7) is admissible. With a=0, θ_S(0,4,7)=t because (0,4,7) is in the support, while θ_S(0,7,4)=1 because (0,7,4) is not. Condition (iii) requires θ(0,4,7)=−θ(0,7,4), i.e. t=1 in the multiplicative group Z3, contradiction. Hence the stated θ_S is not a singular 3-cocycle, and the computation 72+9t is not a valid evaluation of the invariant. This is independent of the unillustrated Figures 5–6; it is a purely algebraic check from the text. The main theorem may still be true, but the paper's only concrete evidence that the invariant has non-trivial power is invalid.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a biquandle-cohomological framework for invariants of oriented immersed surface-links in the four-space, extending the embedded surface-link theory of Carter, Kamada, Saito, and KKKL. It reviews broken surface diagrams and Roseman moves, adds the singular move (h), and proves that the move set (a,b,c,e,f,g,h) is a minimal generating set via the semi-invariant f*. It extends biquandle colorings to diagrams with singular points by imposing the relations a▷b=a▷b and b▷a=b▷a, and it introduces singular 3-cocycles with an extra antisymmetry condition (Definition 4.2(iii)). The main theorem (Theorem 4.5) asserts that the triple-point state-sum is invariant under all generating moves, including (h). The paper ends with an evaluation on the Fenn-Rolfsen link using a claimed singular 3-cocycle θ_S on Z_9, with the computed value 72+9t.","tokens_in":8541,"tokens_out":11787,"duration_ms":117451,"significance":"If Theorem 4.5 holds, the paper provides a plausible and useful extension of biquandle cocycle invariants from embedded to immersed surface-links, and the minimal-generating-set argument via f* is a genuine contribution. The singular cocycle condition is a new algebraic ingredient that could be of independent interest. However, the only concrete demonstration of nontriviality is invalid: the displayed θ_S does not satisfy the manuscript's own Definition 4.2(iii). Since the abstract and introduction advertise the nontrivial value as the evidence that the invariant has strength in the immersed setting, this is a load-bearing gap. The paper also delegates the decisive invariance check for move (h) to figures that are not reproduced in the text as provided, so the main theorem cannot be fully checked from the written argument alone.","major_comments":[{"comment":"The claimed singular 3-cocycle θ_S is not a singular 3-cocycle under the manuscript's own Definition 4.2(iii). In the quandle case used in the example, ▷=∗ and ▷ is first-factor projection, so the singular-compatibility hypothesis becomes b∗c=b and c∗b=c. In the displayed Z_9 table, 4∗7=4 and 7∗4=7, so (b,c)=(4,7) is admissible. Taking a=0, the support of θ_S gives θ_S(0,4,7)=t and θ_S(0,7,4)=1, since (0,7,4) is not in the support. Condition (iii) requires θ_S(0,4,7)=−θ_S(0,7,4), i.e. t=1^{-1}=1, contradicting t^3=1 with t≠1. Thus θ_S violates the singular cocycle condition, and the computation Φ=72+9t does not evaluate the invariant defined in the paper. This invalidates the only explicit evidence of nontriviality.","section":"§4, Example 4.6"},{"comment":"The proof of invariance under move (h) is not checkable from the text as provided. The paper states that Figures 5–6 show the uniqueness of colorings and that condition (iii) makes the Boltzmann weights cancel, but those figures are not reproduced in the text dump and no algebraic verification of the local cases, orientations, or signs is given. Since every local case in these figures is load-bearing for the main theorem, the authors should supply an explicit case-by-case verification or fully labeled figures with the relevant computations written out.","section":"§4, Theorem 4.5 proof"},{"comment":"The independence proof for move (h) uses the fact that f* vanishes for embedded surface diagrams to conclude that f* is invariant under moves (a,b,c,e,f,g) even when the diagram contains singular points elsewhere. This locality step is not justified in the text. It is likely true, but the proof should state explicitly why an embedded Roseman move away from singular points changes f* by the same amount as in a purely embedded diagram; otherwise the semi-invariant argument has a gap.","section":"§2, Proposition 2.2 proof"}],"minor_comments":[{"comment":"The proof refers to 'the claim of Theorem 3.3', but no Theorem 3.3 exists in the manuscript; the reference should be to Proposition 3.3 or Theorem 3.2.","section":"§3, proof of Proposition 3.3"},{"comment":"The convention for the quandle operation table is not stated; the paper should specify whether the entry in row i and column j represents i∗j or j∗i so that the computation of 4∗7 and 7∗4 can be verified unambiguously.","section":"§4, Example 4.6"},{"comment":"The assertion 'Then θ_S is a singular biquandle 3-cocycle' is made without any verification; given that a direct check of condition (iii) fails for the admissible pair (4,7), the authors should include the full cocycle check for any proposed replacement.","section":"§4, Example 4.6"}],"recommendation":"major_revision","confidential_remarks":"The invalid example is the main obstacle: the paper's only demonstration of nontriviality does not hold. I recommend major revision rather than rejection because the general construction may be salvageable and the minimal-generating-set argument is promising, but the authors must provide a valid singular 3-cocycle and a checkable proof of the invariance under move (h). The provided text also omits Figures 5–6, so if they are absent from the submission, the proof of Theorem 4.5 is incomplete."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe takeaway: this paper has a good idea—extending biquandle cocycle invariants to immersed surface-links by adding an antisymmetry condition at singular points—but the only worked example is invalid, so the claimed demonstration of nontriviality does not hold.\n\nWhat is new: Definition 4.2 of a singular 3-cocycle is a natural extension of the KKKL18 biquandle cocycle framework, and the state-sum invariant in Theorem 4.5 is a plausible construction. The coloring-number invariant for immersed surface-links in Proposition 3.3 also looks reasonable. The minimal generating set result, Proposition 2.2, is not new—it already appears in the author's [Jab23]—and the proof here is only a sketch.\n\nWhere it falls down: Example 4.6 is wrong. The asserted theta_S is not a singular 3-cocycle under Definition 4.2(iii). In the quandle case with ▷=* and ▷ first-factor projection, take (a,b,c)=(0,4,7). Then 4*7=4 and 7*4=7, so the singular-compatibility hypothesis holds. But theta_S(0,4,7)=t and theta_S(0,7,4)=1, while condition (iii) requires these to be additive inverses. They are not. Consequently the claimed value 72+9t is not a valid evaluation of the invariant. This is a purely algebraic check; it does not depend on the missing Figures 5–6.\n\nA second soft spot: the invariance proof for move (h) is delegated to Figures 5–6, which are not in the text I have. The argument is plausible—the weight changes sign and the two singular-point arguments swap—but I cannot check the details. Given the example error, I would want a written verification before trusting the main theorem.\n\nWho this is for: low-dimensional topologists working on invariants of immersed surfaces. The core idea could become a useful contribution if the example is fixed or replaced and the move-(h) proof is written out. As it stands, the paper's only concrete evidence of nontriviality is gone.\n\nRecommendation: send it to a serious referee, but expect that the author will need to correct the example or supply a different one, and provide a complete proof for the singular move. The construction deserves another look, but not in this form.","headline":"The paper's core idea is sensible, but the only nontriviality example is algebraically invalid under the paper's own definition, so the demonstration fails.","tokens_in":9029,"tokens_out":5567,"would_cite":false,"duration_ms":47181,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["57K45","57Q35","57R42","57K12"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper defines a state-sum invariant for immersed surface-links in four-space by decorating triple points with weights from biquandle 3-cocycles that satisfy an extra antisymmetry condition at singular points.","keywords":["biquandle cocycle invariant","immersed surface-links","Roseman moves","singular biquandle 3-cocycle","state-sum invariant","triple points","singular points","broken surface diagram"],"falsifier":"Take any broken surface diagram with a singular point, compute the product of Boltzmann weights over all triple points for a biquandle coloring, perform move $(h)$ on a triple point, and recompute: if the two products are not equal for some singular 3-cocycle, Theorem 4.5 is false. A sharper check for the example is to verify directly that the displayed $\\theta_S$ satisfies Definition 4.2: if any quadruple of elements of $\\mathbb{Z}_9$ violates the cocycle identity, or the antisymmetry fails on a pair satisfying the singular relations, the claimed value $72+9t$ does not establish nontriviality.","tokens_in":7983,"feed_emoji":"🕸️","tokens_out":11127,"duration_ms":96553,"temperature":0.7,"pith_summary":"The paper claims that biquandle cocycle invariants, previously defined for embedded surface-links, extend to immersed surface-links in $\\mathbb{R}^4$ provided the 3-cocycles satisfy one extra condition: antisymmetry in the two colors that are identified at a singular point. It proves that the move set $(a,b,c,e,f,g,h)$ generates all equivalences of immersed surface-links and is minimal, so the singular move $(h)$ is genuinely independent of the embedded-case moves. It establishes that biquandle colorings of broken surface diagrams with singular points are in bijection under all these moves, giving a coloring-number invariant, and that the triple-point state-sum with Boltzmann weights is invariant under $(h)$ as well. A worked example link gives the nontrivial value $72+9t$, showing the invariant can detect something in the immersed setting. A reader should care because this supplies an algebraic tool, computable from diagrams, for telling immersed surface-links apart.","feed_headline":"Immersed surface-links gain a cocycle state-sum invariant","feed_subtitle":"Biquandle 3-cocycles with an extra antisymmetry condition survive the singular move, giving computable values.","key_machinery":"The load-bearing object is the singular biquandle 3-cocycle: a biquandle 3-cocycle $\\theta$ with values in an abelian group $A$ that obeys the cocycle identities plus the extra antisymmetry conditions $\\theta(a,b,c)=-\\theta(a,c,b)$ and $\\theta(b,c,a)=-\\theta(c,b,a)$ on pairs of colors satisfying the singular relations. It enters through the triple-point Boltzmann weight: at each triple point $\\tau$ with source-region colors $(a,b,c)$, the weight is $\\theta(a,b,c)^{\\epsilon(\\tau)}$, where $\\epsilon(\\tau)=\\pm1$ is the sign of the triple point, and the state-sum multiplies these weights over all triple points and sums over all colorings. The antisymmetry is exactly the mechanism that makes the two local contributions in move $(h)$ cancel, so this single condition is what carries the extension from embedded to immersed surface-links.","core_discovery":"On the paper's own terms, the central claim is Theorem 4.5: for any singular biquandle 3-cocycle $\\theta$, the state-sum $\\Phi_B^\\theta(L;A)$, computed as $\\sum_C \\prod_{\\tau \\in T(B)} W_B^\\theta(\\tau,C)$ over all biquandle colorings of any broken diagram of an immersed surface-link $L$, is an invariant of $L$. A singular 3-cocycle is a biquandle 3-cocycle with the additional antisymmetry $\\theta(a,b,c)=-\\theta(a,c,b)$ and $\\theta(b,c,a)=-\\theta(c,b,a)$ whenever the two biquandle operations agree on the relevant pair of colors, which is exactly the relation forced at a singular point. The paper further proves that $(a,b,c,e,f,g,h)$ is a minimal generating set of moves for immersed surface-links, and it computes the invariant for an explicit example as $72+9t$, establishing that the invariant can be nontrivial.","pith_inferences":["If Theorem 4.5 is right, the same antisymmetry condition could be imported into other diagrammatic theories with singular points, such as virtual or welded surface diagrams, where move sets of the same shape appear.","The minimality of $(h)$ suggests the state-sum could serve as a singularity detector: one might look for immersed surface-links with identical biquandle colorings but different state-sum values for some cocycle, which would show the invariant sees the immersion itself, not just the underlying coloring data.","The computation through resolutions of a singular marked graph diagram hints at a practical algorithm: read the contributions from type III moves in movie presentations, so larger examples could be computed without drawing full broken-surface pictures.","A fully written case-by-case verification of move $(h)$ would remove the present reliance on the two figures; that is the most direct next step a reader could take."],"forward_implications":["The move set $(a,b,c,e,f,g,h)$ is minimal for immersed surface-links, so the singular move $(h)$ cannot be derived from the embedded moves.","For every finite biquandle, the coloring number $\\#\\mathrm{Col}^B_X(L)$ is an invariant, and the coloring sets of two diagrams of the same immersed link are in bijection.","Every singular biquandle 3-cocycle yields a state-sum invariant, so there is a family of invariants indexed by cocycles and finite biquandles.","The invariant takes a nonconstant value ($72+9t$) on an explicit example, demonstrating that the construction is nontrivial in the immersed setting.","Embedded surface-links are a special case, so the new invariants extend the known biquandle cocycle invariants to a larger class and reduce to them when no singular points are present."],"supporting_citations":[{"why":"Defines biquandle cohomology, the 3-cocycle conditions, the coloring-set bijection, and the embedded state-sum invariant that this paper extends.","marker":"[KKKL18]"},{"why":"Introduces the embedded surface-link moves (a,b,c,e,f,g) that the generating set extends.","marker":"[Ros98]"},{"why":"Supplies the singular move (h) and the ambient move statement for immersed surface-links.","marker":"[AMW17]"},{"why":"Proves the embedded move set (a,b,c,e,f,g) is minimal, which Proposition 2.2 extends by showing (h) is independent.","marker":"[Kaw15]"},{"why":"Provides the surface biquandle coloring framework invoked in the coloring-set bijection theorem.","marker":"[Car09]"},{"why":"Gives the relation $f^*=0$ for embedded broken surface diagrams used in the minimality argument.","marker":"[CS97]"},{"why":"Supplies the elementary proof of $f^*=0$ and the lifting arguments behind the signed triple-point and branch-point count.","marker":"[Sat00]"},{"why":"Provides the example immersed surface-link and its quandle presentation used in the nontriviality computation.","marker":"[Jab23]"},{"why":"Provides the movie-move background and the resolution pictures used to read off type III contributions in the example.","marker":"[CKS04]"}],"fun_headline_variants":["Singular biquandle cocycles yield surface-link invariant","New state-sum invariant for immersed surface-links","Biquandle cocycle invariant survives singular moves","Immersed surface-links: cocycle state-sum with antisymmetry","Minimal moves enable invariant for singular surface-links"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the two hand-drawn local pictures for move $(h)$ cover every case and that, in those pictures, the antisymmetry condition makes the Boltzmann weights cancel; the text asserts this without a written derivation, so if any local case or orientation is wrong the main theorem fails. The example's nontrivial value also assumes the displayed $\\theta_S$ really is a singular 3-cocycle and that the coloring count is 81.","fun_headline_variants_meta":{"raw":{"variants":["Singular biquandle cocycles yield surface-link invariant","New state-sum invariant for immersed surface-links","Biquandle cocycle invariant survives singular moves","Immersed surface-links: cocycle state-sum with antisymmetry","Minimal moves enable invariant for singular surface-links"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000188,"raw_usage":{"total_tokens":1365,"prompt_tokens":1008,"completion_tokens":357,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":624,"completion_tokens_details":{"reasoning_tokens":274}},"tokens_in":624,"tokens_out":357,"duration_ms":3529,"temperature":1.0,"reasoning_tokens":274,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T17:46:37.399233+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take any broken surface diagram with a singular point, compute the product of Boltzmann weights over all triple points for a biquandle coloring, perform move $(h)$ on a triple point, and recompute: if the two products are not equal for some singular 3-cocycle, Theorem 4.5 is false. A sharper check for the example is to verify directly that the displayed $\\theta_S$ satisfies Definition 4.2: if any quadruple of elements of $\\mathbb{Z}_9$ violates the cocycle identity, or the antisymmetry fails on a pair satisfying the singular relations, the claimed value $72+9t$ does not establish nontriviality.","supporting_citations":[],"review_version":2}