{"id":"e59085c9-c3ea-4cc7-ab9c-cc53b36d68db","arxiv_id":"2607.25302","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Rim surgery and all local annulus replacements leave the induced Khovanov homology map of a surface in B^4 unchanged.","lead":"Rim surgery on a surface in the 4-ball does not change the map the surface induces on Khovanov homology, answering a question raised by Hayden and Sundberg. The proof uses Khovanov skein lasagna modules and yields new exotic pairs of surfaces that cannot be related by local annulus replacements.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"First proof's disk-existence claim is false for non-slice γ; full theorem also depends on unpublished functoriality","rationale":"The paper's central claim is plausible and the second proof strategy is elegant, but the manuscript contains a false topological assertion in the first proof (§2) and depends on unpublished work for the full coefficient statement. The false assertion is not merely a missing detail: null-homotopy in B^4 does not guarantee a smooth embedded bounding disk (the trefoil is a counterexample). Since the first proof uses this disk to set up the local model, that proof is invalid as written. The authors' remark that the reduction 'may not be necessary' does not repair the proof because no alternative is given. The second proof, which does not use the disk, could prove the full theorem if the cited functoriality in [Ren26, Ch. 4] is correct; however, that thesis is not publicly available/peer-reviewed, and the published preprint [Ren+25] only covers Q-coefficients. Thus the strongest claim over arbitrary rings rests on an unverified dependency. I therefore recommend CONDITIONAL acceptance: require the authors to fix or remove the false topological step, and to make the functoriality argument for arbitrary rings publicly available. The mathematical core is likely correct, but the paper as written is not self-contained and contains a real error.","tokens_in":8750,"tokens_out":18297,"duration_ms":168765,"concrete_test":"Take Σ to be a Seifert surface for the trefoil knot, embedded in the interior of B^4, and let γ ⊂ Σ be a simple closed curve parallel to the boundary, so γ is isotopic to the trefoil. Check whether there exists a smooth embedded disk D ⊂ int(B^4) with ∂D = γ. Since the trefoil is not slice, no such D exists; this directly disproves the assertion in §2. For the second proof, access [Ren26, Ch. 4] and verify the functoriality of the 'mul' bounded-above Rozansky–Willis homology over Z/2; if the proof uses a hypothesis that fails over Z/2, Theorem 1 lacks support in characteristic 2.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"In §2, Step 2, the authors assert: 'Since γ is contractible in B^4, we may find a smoothly embedded oriented disk D ⊂ int(B^4) bounding γ.' This is false. Null-homotopy does not imply a smooth embedded null-cobordism in dimension 4. A knot K such as the trefoil is null-homotopic in B^4 (π_1(B^4)=0) but is not slice, so it bounds no smoothly embedded disk. A curve γ on a properly embedded surface Σ can be chosen to represent such a K (e.g., a parallel copy of the boundary of a Seifert surface pushed into the interior), so the asserted D need not exist. The subsequent reduction to the model (H_{2,n}, vertical disks, spun annulus) therefore lacks justification. The authors claim this reduction 'may not be necessary,' but no alternative proof of the rim-surgery case is supplied, so the first proof is incomplete. Independently, the second proof over arbitrary commutative rings depends entirely on unpublished functoriality results in [Ren26, Ch. 4] (and the preprint [Ren+25] only proves Q-coefficients); if those are flawed, Theorem 1 over Z/2 or Z is unproven. The topological error is concrete and falsifiable; the thesis dependency is the principal risk to the full-strength claim.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves that rim surgery and, more generally, local annulus replacements on a smooth, oriented, properly embedded surface in B^4 do not change the induced map on Khovanov homology from the empty link to the boundary link. Theorem 1 states this for every commutative ring R. Two proofs are offered. The first uses 0-dimensional-input Khovanov skein lasagna modules, works over a field of characteristic not 2, and relies on a computation from Manolescu–Walker–Wedrich. The second uses 1-dimensional-input lasagna modules and Rozansky–Willis homology; this is the proof that extends to arbitrary coefficient rings. The authors also derive a corollary, using work of Hayden–Sundberg, producing exotic pairs of surfaces in B^4 that cannot be related by any sequence of local annulus replacements.","tokens_in":9018,"tokens_out":9409,"duration_ms":96406,"significance":"If the theorem is correct, it answers a question raised by Hayden and Sundberg and demonstrates that Khovanov homology, via skein lasagna modules, has a new insensitivity property for surface surgeries in the 4-ball. The corollary distinguishing exotic pairs by their non-relation under local annulus replacements is a nice application. The paper is clearly organized and the second proof is attractively short, conditional on substantial machinery from the first author's thesis. However, the first proof contains a concrete false topological assertion, and the full-strength arbitrary-ring statement depends on unpublished thesis material. These issues prevent acceptance in the current form.","major_comments":[{"comment":"The assertion 'Since γ is contractible in B^4, we may find a smoothly embedded oriented disk D⊂int(B^4) bounding γ' is false. Null-homotopy does not imply the existence of a smooth embedded null-cobordism in dimension 4. For example, let K⊂S^3 be a smoothly non-slice knot (e.g., the trefoil), let Σ be a Seifert surface for K pushed into B^4, and let γ be a parallel copy of K in int(Σ). Then γ is null-homotopic in B^4, but if it bounded a smooth disk D, the annulus in Σ between γ and K would make K slice, a contradiction. Thus the topological reduction to the model (H_{2,n}, vertical disks, spun annulus) is unjustified. Since this reduction is what makes the first proof work, the first proof as written is incomplete.","section":"§2, Step 2"},{"comment":"Even if a disk D bounding γ existed, the argument that D can be chosen so that conditions (i) and (ii) hold is too compressed. The obstructions are said to 'take values in Z' and to be controllable by boundary twists, but no definitions or verifications are given. The sentence 'By the even assumption ... the two obstructions sum to an even integer' does not by itself show that both obstructions can be made to vanish simultaneously; the signs of the effects of a boundary twist on the two obstructions need to be tracked explicitly. This is load-bearing for the same reduction as the previous comment.","section":"§2, Step 2 (framing conditions)"},{"comment":"The proof for arbitrary commutative rings depends entirely on functoriality results stated to be in the unpublished PhD thesis [Ren26, Chapter 4]. The cited preprint [Ren+25] is said to establish only Q-coefficient functoriality; the passage to all rings, and indeed to chain-level homotopy functoriality, is attributed to the thesis, which is not available to the reader. If those results are not public, the universal-coefficient statement in Theorem 1 is not verified by this paper. The authors should either include the necessary statements and proofs, or state Theorem 1 with a coefficient restriction that is actually supported by the public literature.","section":"§3, proof of Theorem 1 over arbitrary R"}],"minor_comments":[{"comment":"The phrase 'by linearity, it suffices to prove Theorem 1 when Σ′ is obtained from Σ by a rim surgery (or indeed by a rim surgery with pattern the trefoil knot)' is unclear as it stands. The fact that every annular skein represents a class of the form (1,m(A)) does not, by itself, reduce the general annulus replacement to a single pattern. Since Step 2 later proves the rim-surgery case for every knot K via crossing changes, a more explicit statement of how the reduction is completed would help.","section":"§2, Step 1"},{"comment":"The notation KhR^{-,0,0}_{2,0-div}(1_2) is introduced without a definition of the two superscript triples or the '0-div' subscript. The remark about suppressing 'mul' is helpful, but a short explicit definition of the grading convention would make the paper more readable.","section":"§3, notation"},{"comment":"The dependency on [Ren26] is significant; since the thesis is not yet publicly available, the authors should state which specific results are quoted from it (e.g., Propositions/Theorems), or provide a public preprint version of the relevant chapter.","section":"References"},{"comment":"The description around the standard diagram of 1_4 would benefit from a more detailed explanation of how the orientations alternate and how the 'dotted and undotted caps' are chosen; Figure 2 is referenced but the surrounding text is terse.","section":"§4, Figure 2"}],"recommendation":"major_revision","confidential_remarks":"The first proof's Step 2 contains a false disk-existence claim that directly affects the proof of Theorem 1 over fields of characteristic not 2. The second proof may be correct but is conditional on substantial unpublished material. I recommend major revision rather than rejection because the central idea may be salvageable: the second proof is short and, once the thesis functoriality is available or included, could establish the theorem. However, acceptance in the current form is not advisable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Honest take: this is a real result — rim surgery and local annulus replacements do not change the Khovanov cobordism map, answering Hayden-Sundberg's question, and the corollary about exotic surfaces not related by local annulus replacements is a nice consequence. The second proof, using 1-dimensional-input lasagna modules, is genuinely elegant: the rank-one computation via Hochschild homology of the arc algebra is clean, and the cap-off argument is convincing if you accept the functoriality from Ren's thesis and [Ren+25].\n\nBut there are soft spots. The first proof's Step 2 asserts that because γ is contractible in B^4, it bounds a smoothly embedded disk D. That is false: a knot can be null-homotopic in B^4 without being slice (the trefoil pushed into the interior is a counterexample). The later \"topological reduction\" depends on D, and while the authors say it 'may not be necessary,' they never supply the alternative argument for the rim-surgery case. So as written, the first proof is incomplete. That matters less than it could because the second proof is independent, but it is still a genuine gap in one of the two advertised proofs.\n\nThe bigger risk is the second proof's dependence on unpublished material. The functoriality of Rozansky-Willis homology over arbitrary commutative rings is cited from [Ren26, Chapter 4], a thesis that is not yet available. [Ren+25] only proves Q-coefficients. If that thesis result is flawed, the theorem over Z or Z/2 is unproven. That's not a defect in the mathematics per se, but it makes the current version hard to verify.\n\nThe citation pattern is otherwise fine: [MWW23] is a published computation, and the use of Grigsby-Licata-Wehrli for the symmetric group action is appropriate. No circularity or parameter fitting is present.\n\nVerdict: the theorem is likely true, and the second proof is a promising route, but the manuscript needs a fix to the disk claim and a way to make the thesis dependency public before I'd trust the full-strength statement. Still, it deserves serious refereeing — the question is important and the method is new. I'd send it out, with a referee specifically asked to check the disk existence step and to verify [Ren26].","headline":"Answers Hayden-Sundberg's question with a likely-true theorem, but the first proof contains a false disk-existence claim and the full-strength statement depends on an unpublished thesis.","tokens_in":9493,"tokens_out":3310,"would_cite":false,"duration_ms":34155,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["57K18","57K40"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that rim surgery, and any local annulus replacement, leaves the map induced by a surface on Khovanov homology unchanged over every coefficient ring.","keywords":["rim surgery","local annulus replacement","Khovanov homology","surface cobordism maps","Khovanov skein lasagna modules","Rozansky–Willis homology","exotic surfaces in the 4-ball"],"falsifier":"Over Z/2, take a genus-1 surface in B^4 bounding the unknot, perform a rim surgery along a null-homologous curve with the right-handed trefoil pattern using the even framing, and compute the two induced maps Kh(∅;Z/2) → Kh(L;Z/2). The paper predicts they are equal; any difference is a counterexample to Theorem 1. Alternatively, a gap in the unpublished functoriality proof for Rozansky–Willis homology over Z would remove support for the universal-coefficient claim.","tokens_in":8624,"feed_emoji":"🔗","tokens_out":12668,"duration_ms":116501,"temperature":0.7,"pith_summary":"This paper proves a blindness theorem for Khovanov homology, the link invariant that also assigns a map to every properly embedded surface in the 4-ball. The theorem says that if two such surfaces share a boundary link and are related by a local annulus replacement — cutting out a neighborhood of a curve shaped like S^1×B^1 inside S^1×B^3 and regluing any annulus with the same binding, with rim surgery as the main example — then the two induced maps on Khovanov homology agree, over every commutative coefficient ring. The proof factors both maps through the Khovanov skein lasagna module of the tube S^1×B^3, where a cap-off argument shows that every annulus filling represents the same class; in the one-dimensional-input theory the relevant graded piece is just R. Since rim surgery is a standard tool for building potentially exotic surface pairs, the result identifies a precise limit of Khovanov homology as a detector, and it yields exotic pairs in the 4-ball that cannot be connected by any sequence of local annulus replacements. Two proofs are given: a longer field-by-field argument using 0-dimensional-input skein lasagna modules, and a short universal one using 1-dimensional-input modules built from Rozansky–Willis homology.","feed_headline":"Khovanov homology can't see rim surgery","feed_subtitle":"Two proofs show local annulus replacements leave the induced surface map unchanged—and thus can't reveal exotic surfaces in the 4-ball.","key_machinery":"The load-bearing object is the Khovanov skein lasagna module, a 4-dimensional refinement of Khovanov homology that assigns bigraded groups to pairs (X,L) of a 4-manifold and a framed link in its boundary. The first proof uses the original 0-dimensional-input version over a field of characteristic not 2, and the second uses the 1-dimensional-input version whose input theory is Rozansky–Willis homology for links in connected sums of S^1×S^2. The decisive computation is that the relevant graded piece S^{2,1d,0-div}_{0,0,0}(S^1×B^3; 1_2) — where 1_2 is the standardly framed two-component link S^1×∂B^1 in S^1×S^2 — is isomorphic to R, free of rank one. A cap-off argument in S^4 then shows that ev","core_discovery":"On the paper's own terms, the central claim is Theorem 1: if Σ and Σ' are smooth, oriented, properly embedded surfaces in B^4 with common boundary L ⊂ S^3, and Σ' is obtained from Σ by replacing, in a neighborhood of a curve γ, the standard annulus S^1×B^1 with any smooth annulus A having the same boundary S^1×∂B^1, then for every commutative ring R the maps R ≅ Kh(∅;R) → Kh(L;R) induced by the two surfaces are equal. Rim surgery is a special case, so the theorem answers the open question whether Khovanov cobordism maps can detect rim-surgery exotica: they cannot. The reason is that both maps factor through the skein lasagna module of S^1×B^3 relative to the standard two-component link in it","pith_inferences":["Not in the paper: the same cap-off mechanism suggests any link homology theory whose skein module over S^1×B^3 has a rank-one relevant piece will be blind to local annulus replacements; testing this with sl(N) Khovanov–Rozansky homology is a direct next step.","The theorem indicates that Khovanov-homology detectability of exotic surfaces in the 4-ball is non-local: any invariant that does detect rim-surgery exotica must see the global embedding, not just the surface near a curve.","A reader who wants to rely on the arbitrary-ring version should first verify the cited functoriality of Rozansky–Willis homology in the unpublished thesis; until then, only the characteristic-not-2 field version stands on the first proof alone."],"forward_implications":["Khovanov cobordism maps cannot distinguish surfaces in B^4 that differ by a local annulus replacement, so rim surgery is invisible to this invariant over every coefficient ring.","For every genus g ≥ 0 there exist exotic surface pairs in B^4, with common boundary a knot, that are not related by any sequence of local annulus replacements or rim surgeries.","The insensitivity extends to any simply connected 4-manifold: two surfaces related by local annulus replacements represent the same element of the skein lasagna module.","Taking parallel cables does not restore detectability: for any n, the induced Khovanov maps on the n-cables of the two surfaces also agree.","The equality of maps holds for every commutative ring R, including Z/2, where the first proof's field assumption fails."],"fun_headline_variants":["Rim surgery invisible to Khovanov homology","Two proofs: rim surgery leaves Khovanov map fixed","Khovanov maps ignore rim surgery and annulus swaps","Rim surgery can't alter Khovanov cobordism map","Khovanov homology blind to rim-surgery exotica"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is that Rozansky–Willis homology is functorial over every commutative ring, a result cited from an unpublished thesis; if that functoriality fails, the arbitrary-ring version of the theorem collapses (the first proof covers only fields of characteristic not 2).","fun_headline_variants_meta":{"raw":{"variants":["Rim surgery invisible to Khovanov homology","Two proofs: rim surgery leaves Khovanov map fixed","Khovanov maps ignore rim surgery and annulus swaps","Rim surgery can't alter Khovanov cobordism map","Khovanov homology blind to rim-surgery exotica"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000205,"raw_usage":{"total_tokens":1193,"prompt_tokens":668,"completion_tokens":525,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":412,"completion_tokens_details":{"reasoning_tokens":443}},"tokens_in":412,"tokens_out":525,"duration_ms":5830,"temperature":1.0,"reasoning_tokens":443,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T02:48:46.861443+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Over Z/2, take a genus-1 surface in B^4 bounding the unknot, perform a rim surgery along a null-homologous curve with the right-handed trefoil pattern using the even framing, and compute the two induced maps Kh(∅;Z/2) → Kh(L;Z/2). The paper predicts they are equal; any difference is a counterexample to Theorem 1. Alternatively, a gap in the unpublished functoriality proof for Rozansky–Willis homology over Z would remove support for the universal-coefficient claim.","supporting_citations":[],"review_version":1}