{"id":"576b1a5e-ca09-418c-a09d-c9e380d0b94a","arxiv_id":"2608.07114","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"An improved tangle algorithm computes involutive Khovanov invariants and shows the Whitehead doubles of two pretzel knots carry an equivariant Rasmussen invariant (0,2), yielding new exotic slice disk pairs.","lead":"This paper introduces a faster, tangle-based way to compute a symmetry-enhanced knot invariant, and uses it to identify new examples where a slice disk cannot be smoothly deformed into its rotated copy. The results give concrete evidence about old open questions on Whitehead doubles and exotic slice disks in four dimensions.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The new P(-5,5,-5) case rests entirely on an unarchived 17-hour yui run for Wh(P(-5,5,-5)); a bug in the implementation of Algorithm 4.12 or in the linear-system readout would remove the only novel example.","rationale":"The central theorem's logical structure is sound: assuming Proposition 1.7 and Theorem 4, the obstruction theorem [San25, Corollary 1.11] and the Z-slice disk topological isotopy results imply the exotic-pair conclusion. The tangle framework is a genuine extension of Bar-Natan's formalism, and the p=3 hand computation in Section 6 is reasonably detailed. The weakest point is precisely the unverified, unarchived computation for Wh(P(-5,5,-5)), which the reader also identified. Because that computation is the only evidence for the novel P(-5,5,-5) case, it is load-bearing: any implementation error in Algorithm 4.12 or in the linear-system readout would invalidate Proposition 1.7 and hence the new example. The secondary gap in the general-p proof of Theorem 4 is real but less severe, since the p=3 derivation illustrates the method and the p=5 case could be checked by the same diagrammatic computation. No internal inconsistency was found in the theoretical sections; the appropriate verdict remains conditional pending reproducible computational artifacts and an explicit p=5 check.","tokens_in":27671,"tokens_out":6137,"duration_ms":58067,"concrete_test":"Pin yui to a specific commit, publish the exact PD-code for Wh(P(-5,5,-5)), rerun the 17-hour computation, and confirm that the output is (s,s)=(0,2); also publish logs and checksums. Independently recompute the h-divisibilities of the two Lee-class images z0,z1 from the reduced complex using a second implementation of Algorithm 4.12, or verify by hand the reduction for a smaller analogous diagram such as Wh(P(-3,3,-3)). If the rerun or the independent check disagrees, Proposition 1.7 fails and the P(-5,5,-5) part of Theorem 1 is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The novel half of Theorem 1 is the P(-5,5,-5) case, and its only source of evidence is Proposition 1.7: the computation reporting equivariant Rasmussen invariant (0,2) for Wh(P(-5,5,-5)). The paper describes this computation in one sentence (72-crossing diagram, 17 hours on 64 cores with 3 TB memory) and gives no logs, no commit hash, no output data, and no independent verification. The computation exercises the new Algorithm 4.12 and the linear-system readout of Corollary 5.3; a coding error in, for example, the equivariant Gaussian elimination cases of Proposition 4.9, the symmetry-breaking reduction of Proposition 4.11, or the extraction of h-divisibilities would directly corrupt the reported invariant. If the readout is wrong, the conclusion that Wh(P(-5,5,-5)) is not isotopy-equivariantly slice has no support, and with it the exotic-pair conclusion for the P(-5,5,-5) case disappears; the P(-3,3,-3) case is already covered by prior work. A secondary concern is that Theorem 4 is proved in detail only for p=3, with the statement 'the proof for the general case proceeds similarly'; since Theorem 1 needs p=5, that case should also be checked explicitly rather than inferred from the p=3 diagram. The theoretical framework is coherent and the p=3 computation is detailed, but the single computational data point for the new example is the most load-bearing and least verified input.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper extends Bar-Natan's tangle-categorical framework to an involutive setting, defining τ-equivariant Khovanov complexes for involutive tangles and a formal involutive complex Q(L) for involutive links. It proves invariance under involutive Reidemeister moves and the I-move, develops equivariant versions of delooping and Gaussian elimination, and packages these into a divide-and-conquer algorithm (Algorithm 4.12) with a linear-system criterion for the equivariant Rasmussen invariant (Corollary 5.3). The paper then states two main computational results: Theorem 4 asserts (s, s) = (0,2) for all odd p ≥ 3 pretzel knots P(−p, p, −p), with a detailed proof only for p=3; Proposition 1.7 asserts that the Whitehead doubles of P(−3,3,−3) and P(−5,5,−5) have equivariant Rasmussen invariant (0,2), the second obtained from a single 17-hour run of the program yui on a 72-crossing diagram. Combining these with known results yields Theorem 1: those knots and their Whitehead doubles are not isotopy-equivariantly slice, and the Whitehead doubles admit exotic pairs of slice disks. The P(−3,3,−3) case is already covered by prior work, so the novel content is the P(−5,5,−5) case.","tokens_in":27943,"tokens_out":5794,"duration_ms":54834,"significance":"If the computational claims are correct, the paper provides a substantial algorithmic advance: the tangle-based reduction appears to convert previously infeasible involutive Khovanov computations into fast ones, and the method for extracting h-divisibilities from linear systems is clearly useful. The paper also explicitly repairs the missing I-move in the author's earlier [San25], which is a genuine foundational correction. The hand computation for p=3 in Section 6 is a valuable demonstration of the equivariant diagrammatic method. However, the significance is tempered by the fact that the only new example, the P(−5,5,−5) Whitehehead double, rests entirely on an unarchived computational run, and the general-p statement of Theorem 4 is not actually proved beyond p=3. The paper does not ship machine-checked proofs or reproducible computational artifacts; the code is mentioned but no version, logs, or output data are supplied.","major_comments":[{"comment":"The P(−5,5,−5) case of Theorem 1 is supported only by a one-sentence description of a 17-hour run of yui on a 72-crossing diagram, with no commit hash, input file, logs, or output data. The reported invariant (0,2) is produced by the new Algorithm 4.12 and the linear-system readout of Corollary 5.3, so an implementation error in the equivariant Gaussian elimination of Proposition 4.9, the symmetry-breaking reduction of Proposition 4.11, or the extraction of h-divisibilities would directly corrupt the result. Since the P(−3,3,−3) case is already covered by [DMT26], the entire novelty of Theorem 1 depends on this unverified run. The authors should provide an auditable record, including the exact yui version, the input diagram in machine-readable form, the computed output, and ideally an independent verification, before the P(−5,5,−5) conclusion can be accepted.","section":"§1, Proposition 1.7"},{"comment":"The proof of Theorem 4 is carried out in detail only for p=3, with the final sentence stating that the general case proceeds similarly. This is not adequate for the theorem as stated, and Theorem 1 specifically needs p=5. The reduction in Section 6 uses complexes E^+−3, E^−3 with specific length and differential; for p=5 these complexes have more summands and potentially different cancellation patterns, so the claim that the same obstruction produces an h-factor of exactly one requires argument. The paper should either present an explicit p=5 computation, prove a uniformity lemma showing that the reductions and cancellations are identical in form for all odd p, or restrict Theorem 4 to p=3 and state the remaining cases as a conjecture.","section":"§6, Theorem 4"}],"minor_comments":[{"comment":"Remark 3.14 concedes that [·]τ is not shown to descend to the equivariant movie-move quotient; the introduction and Theorem 2 are worded as if full functoriality on involutive tangles is established, so the paper should either prove this or clearly state which claims are provisional.","section":"§3.3, Remark 3.14"},{"comment":"Remark 3.17 states that the τ-equivariant lift [S]τ is defined only after choosing an isotopy from S to τS and that uniqueness up to homotopy is not established. The text should explicitly say whether any of the main results, in particular the computations of Section 6, depend on this choice; if they do not, that should be stated.","section":"§3.3, Remark 3.17"},{"comment":"The correction of the published [San25, Proposition 3.3] is mathematically important, since the τ-invariance of the equivariant Lee cycle is used in the definition of the equivariant Rasmussen invariant; the corrected statement and its proof should appear in the main text rather than only in a footnote.","section":"Footnote 4"},{"comment":"The notation for the sequential complexes E^+−3 and E^−3 is garbled in the displayed text (the lines ending in '= a b s' and '= s b a' do not render the intended arrows); please provide a properly typeset version of these complexes.","section":"§6, displayed complexes"},{"comment":"The equality e − Iτe′ = e(I − Iτ) = 0 in the proof of Claim 6.1 would be easier to verify if the action of Iτ on the relevant A-end objects were shown in a small diagram, since this is a key step in establishing the claim.","section":"§6, Claim 6.1"}],"recommendation":"major_revision","confidential_remarks":"The main obstacle to acceptance is computational reproducibility: the sole new example rests on an unarchived 17-hour run, and the general-p theorem is not proved beyond p=3. Both issues are fixable within the manuscript's scope by providing archival data, an explicit p=5 computation or uniformity argument, and clarifications of the limitations already acknowledged in Remarks 3.14 and 3.17. I would not reject, but I would not accept without these additions."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my take. This paper does something genuinely useful: it extends involutive Khovanov homology to tangles and gives equivariant delooping, Gaussian elimination, and a symmetry-breaking reduction that make the invariant far more computable. The categorical invariance arguments are careful, and the paper explicitly fixes the missing I-move from [San25]. The hand proof of Theorem 4 — that P(-p,p,-p) has equivariant Rasmussen invariant (0,2) for every odd p — is new and does not depend on the big computer run. The speedup is real, not cosmetic: one example drops from 10 minutes to 200 ms. If the computational claims are right, the exotic slice disk pair for Wh(P(-5,5,-5)) is a solid new result, and the P(-3,3,-3) case is already covered by [DMT26].\n\nThe soft spot is exactly where the stress test points. Proposition 1.7 reports (0,2) for both Whitehead doubles, but the P(-5,5,-5) value comes from a 72-crossing diagram, 64 cores, 3 TB of memory, 17 hours, and no logs, no commit hash, no output data. Algorithm 4.12 and Corollary 5.3 are new and involved, so a coding error could change the invariant. The P(-3,3,-3) case is already known, so the entire novelty of the P(-5,5,-5) exotic pair rests on that one unverified run. The GitHub repo is public, but no exact version is pinned to the computation.\n\nA smaller gap: Theorem 4 is proved in full for p=3 and then handled by \"the proof for the general case proceeds similarly.\" The pattern is probably clear, but since Theorem 1 needs p=5, that case deserves an explicit check rather than an inference.\n\nNone of this is fatal to the theoretical framework. The mathematics is coherent, the p=3 computation is shown in detail, and the program is honestly described with an AI declaration. The paper deserves serious peer review. My recommendation: send it out, but require reproducible artifacts for the P(-5,5,-5) computation — a pinned commit, logs, and output data at minimum — and ask the author to expand the general-p argument. With those in hand, I would take the exotic pair conclusion as established.","headline":"A genuinely useful tangle extension of involutive Khovanov homology, but the new P(-5,5,-5) exotic slice disk pair rests on an unarchived 17-hour computational run that needs reproducible artifacts before I'd take it as established.","tokens_in":782,"tokens_out":850,"would_cite":true,"duration_ms":37688,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["57K18","57K10"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper extends involutive Khovanov homology to tangles and uses the resulting speedup to prove that the Whitehead doubles of the pretzel knots P(-3,3,-3) and P(-5,5,-5) admit exotic pairs of slice disks.","keywords":["involutive Khovanov homology","equivariant Rasmussen invariant","strongly invertible knots","slice disks","exotic pairs","tangle Khovanov homology","pretzel knots","Whitehead doubles"],"falsifier":"Recompute the equivariant Rasmussen invariant of $\\mathrm{Wh}(P(-5,5,-5))$ with an independent implementation or with released logs from the program and obtain a value other than $(0,2)$, or exhibit a smooth isotopy rel boundary between $\\mathrm{Wh}(D)$ and $\\tau\\mathrm{Wh}(D)$ for some slice disk $D$ of $P(-5,5,-5)$.","tokens_in":27418,"feed_emoji":"🧶","tokens_out":12772,"duration_ms":107928,"temperature":0.7,"pith_summary":"The paper aims to make involutive Khovanov homology and the associated equivariant Rasmussen invariant computable on large knots by extending the tangle-and-cobordism formulation of Khovanov homology to involutive tangles and reducing the computation piecewise. The payoff is a concrete 4-dimensional result: for the strongly invertible slice pretzel knots $K=P(-3,3,-3)$ and $K=P(-5,5,-5)$, no slice disk is smoothly isotopic rel boundary to its symmetric counterpart, and after Whitehead doubling the two slice disks $\\mathrm{Wh}(D)$ and $\\tau\\mathrm{Wh}(D)$ are topologically isotopic but not smoothly isotopic, forming an exotic pair. A sympathetic reader should care because this is the first computation of such invariants at a 72-crossing scale, it gives a new example not previously known, and it turns a proposed method for detecting equivariant sliceness into a practical algorithmic tool.","feed_headline":"Two pretzel knots yield exotic pairs of slice disks","feed_subtitle":"Involutive Khovanov homology, computed tangle-by-tangle, separates slice disks of Whitehead doubles that are topologically the same.","key_machinery":"The central object is the $\\tau$-equivariant Khovanov complex $[T]_\\tau$ of an involutive tangle $T$, namely the formal tangle Khovanov complex $[T]$ together with a canonical isomorphism $i:[T]\\to\\tau[T]$ encoding the symmetry. Invariance is proved under the involutive Reidemeister moves and the $I$-move, a non-local move required when the ambient space is $S^3$ rather than $\\mathbb{R}^3$, and taking the cone of $I-I_\\tau$ recovers the earlier involutive Khovanov complex. The computational engine is a reduction pipeline: a symmetric tangle decomposition $L=D(T_s,T_a,\\tau T_a)$, equivariant delooping and Gaussian elimination on the on-axis piece, ordinary delooping and Gaussian elimination on the off-axis piece, assembly, cone formation, symmetry-breaking elimination, and finally a Rees-correspondence linear system that reads off the $h$-divisibility of equivariant Lee cycles without computing full homology over $\\mathbb{F}_2[h]$. This pipeline is what carries both the hand computation for pretzel knots and the 72-crossing machine computation.","core_discovery":"On the paper's own terms, the central discovery is that the equivariant Rasmussen invariant $(\\underline{s},\\bar{s})$ of a strongly invertible link can be obtained from a tangle-by-tangle decomposition of the involutive Khovanov complex, and that this makes the invariant strong enough to separate smoothly non-isotopic slice disks that are topologically isotopic. For $K=P(-3,3,-3)$ and $P(-5,5,-5)$, the paper computes $(\\underline{s},\\bar{s})=(0,2)$ for $K$ itself, and reports the same value $(0,2)$ for the Whitehead doubles, the latter for a 72-crossing diagram. Because a non-trivial equivariant Rasmussen invariant obstructs isotopy-equivariant sliceness, $K$ and $\\mathrm{Wh}(K)$ are not isotopy-equivariantly slice; combining this with the fact that any two $\\mathbb{Z}$-slice disks are topologically isotopic rel boundary yields the exotic pair of slice disks for the two Whitehead doubles. The $P(-3,3,-3)$ statement was already known by other means, while the $P(-5,5,-5)$ statement is presented as new.","pith_inferences":["The paper leaves implicit that the same divide-and-conquer scheme should work for other equivariant Lee-class or slice-torus invariants, not just the pair $(\\underline{s},\\bar{s})$.","One could test the method's reach by applying it to the strongly invertible knots with anomalous invariants listed in the paper and looking for further exotic pairs after satellites other than Whitehead doubling.","The only load-bearing computational datum is the reported 17-hour run on $\\mathrm{Wh}(P(-5,5,-5))$; an independent implementation or a released run log would convert that computational evidence into a checkable proof ingredient.","If Question 1.9 holds broadly, the paper's list of knots with anomalous equivariant Rasmussen invariant would generate additional families of exotic slice-disk pairs after Whitehead doubling."],"forward_implications":["The same pipeline should make involutive Khovanov homology and the equivariant Rasmussen invariant routinely computable for strongly invertible knots well beyond the 10-crossing dataset previously screened.","For every odd $p\\ge 3$, the pretzel knot $P(-p,p,-p)$ has invariant $(\\underline{s},\\bar{s})=(0,2)$, so none of these slice pretzel knots is isotopy-equivariantly slice.","The Whitehead doubles of $P(-3,3,-3)$ and $P(-5,5,-5)$ have invariant $(0,2)$ as well, so their slice disks provide explicit exotic pairs in the 4-ball.","These examples give the first positive evidence for the paper's Question 1.9 that a non-trivial equivariant Rasmussen invariant of a strongly invertible slice knot should persist under Whitehead doubling."],"supporting_citations":[{"why":"defines the involutive Khovanov complex and equivariant Rasmussen invariant for knots, and supplies the obstruction corollary that non-trivial invariant rules out isotopy-equivariant sliceness.","marker":"[San25]"},{"why":"supplies the tangle-and-cobordism formalism of Khovanov homology on which the $\\tau$-equivariant complex is built.","marker":"[Bar05]"},{"why":"supplies delooping and Gaussian elimination, adapted equivariantly for the reduction algorithm.","marker":"[Bar07]"},{"why":"gives the diagrammatic Lee-class and $h$-divisibility method used in the pretzel-knot computation.","marker":"[KS25]"},{"why":"introduces the involutive Reidemeister moves used in the invariance theorem for the $\\tau$-equivariant complex.","marker":"[L W21]"},{"why":"introduces the $I$-move needed for involutive isotopy in $S^3$, also used in the invariance theorem.","marker":"[Bor+26b]"},{"why":"provides the fact that $\\mathrm{Wh}(D)$ is a $\\mathbb{Z}$-slice disk of $\\mathrm{Wh}(K)$, used to reach the topological-isotopy conclusion.","marker":"[Gut+23]"},{"why":"supplies the topological-isotopy classification of $\\mathbb{Z}$-slice disks used for the exotic-pair conclusion.","marker":"[CP21]"},{"why":"provides the companion topological-isotopy result for $\\mathbb{Z}$-slice disks cited in the same step.","marker":"[Hay21]"},{"why":"is the improved program used for the 72-crossing computation of the equivariant Rasmussen invariant of $\\mathrm{Wh}(P(-5,5,-5))$.","marker":"[San26]"}],"fun_headline_variants":["Exotic slice disk pairs found for two Whitehead doubles","Involutive Khovanov homology distinguishes exotic slice disks","Tangle-by-tangle invariant reveals exotic slice disks for Whitehead doubles","New proof: Whitehead doubles of pretzel knots have exotic slice disks"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the correctness of the reported 72-crossing computation of $\\mathrm{Wh}(P(-5,5,-5))$ and the obstruction theorem that a non-trivial equivariant Rasmussen invariant forbids isotopy-equivariant sliceness; if either gives way, the $P(-5,5,-5)$ conclusion collapses.","fun_headline_variants_meta":{"raw":{"variants":["Exotic slice disk pairs found for two Whitehead doubles","Involutive Khovanov homology distinguishes exotic slice disks","Tangle-by-tangle invariant reveals exotic slice disks for Whitehead doubles","New proof: Whitehead doubles of pretzel knots have exotic slice disks"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001046,"raw_usage":{"total_tokens":4375,"prompt_tokens":903,"completion_tokens":3472,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":519,"completion_tokens_details":{"reasoning_tokens":3402}},"tokens_in":519,"tokens_out":3472,"duration_ms":22005,"temperature":1.0,"reasoning_tokens":3402,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T14:35:40.302245+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the equivariant Rasmussen invariant of $\\mathrm{Wh}(P(-5,5,-5))$ with an independent implementation or with released logs from the program and obtain a value other than $(0,2)$, or exhibit a smooth isotopy rel boundary between $\\mathrm{Wh}(D)$ and $\\tau\\mathrm{Wh}(D)$ for some slice disk $D$ of $P(-5,5,-5)$.","supporting_citations":[],"review_version":1}