{"id":"6776719a-c73f-4d87-9c78-bb5a84a9df14","arxiv_id":"2507.13642","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The Borel refinement of Bar-Natan Khovanov homology detects that equivariant slice genus can be arbitrarily larger than isotopy-equivariant slice genus.","lead":"This paper builds a new Khovanov-type invariant for knots with a 180-degree rotation symmetry and uses it to show that the equivariant slice genus and the isotopy-equivariant slice genus can differ by an arbitrarily large amount. The result answers an open distinction problem in 4-dimensional knot theory and introduces a suite of invariants for studying equivariant cobordisms.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Deferred genericity results in [BDMS25] are load-bearing for the Borel invariant's well-definedness; if they fail, the esQ-based bound in Theorem 1.3 collapses.","rationale":"The reader's CONDITIONAL verdict is well supported. The paper is internally coherent, with unusually detailed algebraic proofs, explicit diagrams for the M1/M2/M3 invariance, and a concrete mechanism (the Borel construction) that genuinely distinguishes equivariant from isotopy-equivariant data. However, the foundational genericity statements are not merely cosmetic: they underwrite the definition of the invariant itself. The paper openly states that rigorous proofs have not appeared and defers them to [BDMS25]. Since Theorems 1.1 and 1.2 are the bridge from the Borel complex to the genus bound, this is exactly the kind of load-bearing dependency that should gate acceptance. The reader's weakest_assumption identifies this accurately. I did not find a separate internal inconsistency in the main algebraic construction—the Koszul duality and connected-sum formalism are plausible, and the tensor-product computation in Lemma 6.29 is intricate but not obviously flawed. The hand enumeration for J is a real secondary concern, but it is testable and would only affect the specific example, not the framework. Thus the right verdict remains CONDITIONAL until the deferred proofs or an independent verification of the genericity statements is supplied.","tokens_in":63607,"tokens_out":16809,"duration_ms":184963,"concrete_test":"Verify the two deferred statements in [BDMS25]: (1) completeness of the equivariant Reidemeister moves together with the I-move and R-move for Sakuma equivalence; (2) existence of an equivariantly generic movie for every equivariant cobordism after a boundary-moving isotopy. A practical computational check is to enumerate transvergent diagrams up to 10–13 crossings (using [BO25] plus SnapPy PD-code symmetries) and test whether every Sakuma-equivalent pair is connected by the listed moves; separately, enumerate codimension-one singularities in a generic 1-parameter family of equivariant projections and check that they specialize to the moves of Figure 2.5. If either check reveals a missing move or a non-generic perturbation, the invariance proof and hence Theorem 1.3 fail.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that eg4(#mJ) can exceed eig4(#mJ) runs through Theorem 1.1 (invariance of the Borel complex and existence of equivariant cobordism maps), Theorem 1.2 (the esQ genus bound), and Lemma 6.31 (conversion of the algebraic lower bound into a genus bound). All of these presuppose that every involutive link can be represented by a transvergent diagram, that the move set in Theorem 2.10—including the I-move and R-move—is complete for Sakuma equivalence, and that every equivariant cobordism can be equivariantly isotoped into an equivariantly generic movie with elementary moves as in Figure 2.5. The paper explicitly defers these statements to the forthcoming [BDMS25] (Section 2.2 after Definition 2.9; proof of Theorem 2.10; Definition 2.12). If any of them fails, Kc_Q and esQ are not invariants, and the lower bound on eg4(#mJ) is unsupported. This is not a manufactured concern: the authors themselves flag the gap. A separate, secondary soft spot is the hand enumeration in Lemma 6.13/Example 4.17, which is summarized rather than fully documented; an error there would also break the lower bound, but it is more local and testable.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a Bar-Natan homology refinement for involutive links. The main construction is the Borel complex Kc_Q(L) = (Kc^-(L) ⊗ F[Q], ∂ + Q(1+τ)), whose homotopy type is asserted to be invariant under Sakuma equivalence (Theorem 1.1), and which supports cobordism maps for equivariant cobordisms. From the reduced Borel complex the authors define numerical invariants es_Q and es_{Q,A,B} that bound the genus of equivariant cobordisms (Theorem 1.2). The principal application is Theorem 1.3: for the strongly invertible knot J = 17nh74, eg_4(#mJ) ≥ ⌈m/2⌉ while eig_4(#mJ) ≤ 1 for all m, exhibiting an arbitrarily large gap between the equivariant and isotopy-equivariant slice genus. The proof combines a local computation for J (Lemma 6.13), a connected-sum formula for Borel complexes obtained via Koszul duality (Theorem 6.25), a growth calculation for tensor powers (Lemma 6.30), and a topological stabilization argument (Lemma 6.32). The paper also develops mixed complexes to incorporate the Lobb–Watson filtration.","tokens_in":63822,"tokens_out":4578,"duration_ms":56207,"significance":"If the main theorems are correct, this is a substantial contribution: it gives the first proof that equivariant slice genus and isotopy-equivariant slice genus can differ, and that the gap can be arbitrarily large. The invariant is genuinely new in that it records higher homotopy-commutation data of the involution rather than only the homological action, and the paper explicitly identifies why Sano-style mapping-cone invariants, which are functorial for isotopy-equivariant cobordisms, cannot see this separation. The paper is also commendably concrete: it provides a computational program [BO25], detailed small examples, and falsifiable numerical predictions. The main theorems are stated carefully, and the algebraic framework for connected sums via Koszul duality is a useful and natural tool. The significance is, however, conditional on the postponed equivariant genericity statements and on the hand-verified enumeration in the key example.","major_comments":[{"comment":"The well-definedness of the Borel complex and of the equivariant cobordism maps is load-bearing for Theorems 1.1, 1.2, and 1.3, but it depends on deferred genericity statements. The paper explicitly states that rigorous proofs of the existence of transvergent diagrams for involutive links, the completeness of the equivariant Reidemeister move set, including the I-move and R-move, and the existence of equivariantly generic cobordism movies will appear in [BDMS25]. Since Theorem 1.3 cannot hold if Kc_Q is not an invariant, these statements are not peripheral. A complete proof of these genericity results, or a published reference containing them, is required before the central claim can be regarded as established.","section":"Section 2.2 (after Definition 2.9), Theorem 2.10, Definition 2.12"},{"comment":"The invariance of Kc_Q under the M3 move is proved by a mapping-cone argument followed by a reduction to Bar-Natan's tangle category. The final step asserts that the constructed map F is, up to homotopy, the unique map coming from a morphism in Kob(R). This requires checking that the mapping-cone identifications are compatible with the tangle-category functor at each stage, not just that the tangles are simple. The paragraph currently gives only a heuristic justification; please spell out the naturality diagram that identifies the cone of the Borel complexes with the image under Kc^- of the corresponding cone in Kob(R). As written, the M3 invariance proof has a gap.","section":"Section 5.3, M3 move"},{"comment":"The lower bound in Theorem 1.3 depends on the existence of a local map from the complex C_Q of Example 5.9 into Kcr_Q(J). Lemma 6.13 rests on an 'exhaustive analysis' of the possible differential components X_i, Y_i, Z_i satisfying (6.14) and ∂_Q^2 = 0, but the enumeration is summarized rather than fully documented. Similarly, Example 4.17 refers to a 'straightforward but tedious exercise' enumerating extensions of τ. Because a single missed differential component could change the local equivalence class and hence the es_Q invariants, the argument is load-bearing. Please provide a complete case analysis or a machine-checkable verification for the enumeration.","section":"Section 6.3, Lemma 6.13 and Example 4.17"}],"minor_comments":[{"comment":"There is a typo: 'refinemenet' should be 'refinement.'","section":"Section 1.4"},{"comment":"The statement of Lemma 6.28 contains a likely typo in the tensor-product direction: from the proof and from the use in Lemma 6.31, the conclusion should involve a local map Y1 ⊗ Y2 → Y1' ⊗ Y2' (with the appropriate ⊗_B product), not 'Y1 ⊗ Y1' → Y2 ⊗ Y2'.","section":"Section 6.5.3, Lemma 6.28"},{"comment":"In the displayed chain Σ_{i=0}^{⌊m/2⌋} u^{m-i} Q^{2i} x_i, the elements x_i are never defined. Presumably they denote suitable tensor-product generators in (C_Q)^{⊗m}; please define them explicitly, as this is the cycle used to prove the growth of es_Q.","section":"Section 6.6, Lemma 6.30"},{"comment":"The notation 'dKcr_p(D)' and 'dKcr_un(D)' appears to be introduced without definition; it presumably means Kcr_p(D)/(u=0), but this should be stated.","section":"Section 3.2"},{"comment":"Theorem 2.10 refers to moves '(IR-1) through (M-3)' in Figure 2.2, but the figure is not annotated with the move names here. Please label the moves explicitly so that the later references to M1, M2, and M3 in Section 5.3 can be checked.","section":"Section 2.2, Figure 2.2 and Theorem 2.10"}],"recommendation":"major_revision","confidential_remarks":"The largest risk is the deferral of the equivariant genericity results to [BDMS25]; if those results are not available in full, the Borel invariant and the main theorem are unsupported. The hand enumeration in Lemma 6.13 is the second risk and should be independently verified. If both are resolved, the paper is likely a strong contribution. It may be worth asking the authors to provide a proof-of-concept of Lemma 6.13 with reproducible code, possibly extending [BO25], since the rest of the argument is carefully structured."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main theorem is the big thing: for J = 17nh74, eg4(#mJ) >= ceil(m/2) while eig4(#mJ) <= 1, so the two genera really differ and the gap can be arbitrarily large. I think the theorem is correct, with one condition attached.\n\nWhat the paper does well: the Borel construction over the full F[Q] instead of F[Q]/Q^2 is a genuine formal departure from Sano, and it gives access to higher Q-information that the truncated theory loses. The esQ invariants, especially the truncated esQ,A,B, are natural and the connected-sum localization result in Section 6.4 is neat. They put real work into the M1/M2/M3 move invariance, writing down explicit homotopies rather than waving hands. The paper is also honest: it explicitly says that key genericity statements about transvergent diagrams and equivariant cobordism movies are deferred to a forthcoming paper [BDMS25]. That candor is to their credit.\n\nThe soft spots are exactly where the stress-test points. The invariance of the Borel complex, Theorem 1.1, and the existence of equivariant cobordism maps presuppose that every involutive link has a transvergent diagram, that the move set in Theorem 2.10 (including the I-move and R-move) is complete for Sakuma equivalence, and that equivariant cobordisms can be made equivariantly generic. If any of those fails, the esQ invariants are not well-defined and the lower bound in Theorem 1.3 collapses. The authors flag this gap themselves, so it is not a hidden flaw, but it is load-bearing. The secondary soft spot is the enumeration in Lemma 6.13: the possible differential components are listed, but the 'straightforward but tedious' part is summarized. That looks local and checkable, so I am less worried about it.\n\nIf the companion paper delivers the genericity results, this should be accepted. The central argument is coherent, the algebra is detailed, and the examples are convincing. The paper is for people working on equivariant knots and Khovanov-type invariants; it deserves a serious referee, not a desk reject.","headline":"The Borel complex over F[Q] is a real advance and the genus-gap theorem is likely correct; the main risk is the deferred genericity results, which the authors flag.","tokens_in":64413,"tokens_out":2389,"would_cite":true,"duration_ms":27563,"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 proves that the Borel construction on Bar-Natan homology gives invariants that make the equivariant slice genus of $\\#^m J$ grow at least as $\\lceil m/2\\rceil$ while the isotopy-equivariant slice genus stays at most 1.","keywords":["Khovanov homology","Bar-Natan homology","involutive links","strongly invertible knots","equivariant slice genus","Borel construction","equivariant cobordisms","Lobb-Watson filtration"],"falsifier":"Run the enumeration described in Lemma 6.13 on a transvergent diagram of $J=17\\mathrm{nh}74$ and check for a grading-shift-zero local map from the five-generator complex $C_Q$ into $\\widehat{\\mathrm{Kcr}}^-_Q(J)$; absence of such a map would break Lemma 6.13 and the bound $\\mathrm{eg}_4(\\#^mJ)\\ge\\lceil m/2\\rceil$.","tokens_in":63376,"feed_emoji":"🪢","tokens_out":14974,"duration_ms":153196,"temperature":0.7,"pith_summary":"This paper aims to show that two natural notions of 'symmetric slice surface' genuinely differ. The equivariant slice genus $\\mathrm{eg}_4(K)$ asks for a surface in the 4-ball fixed setwise by the standard extension of the involution, while the isotopy-equivariant slice genus $\\mathrm{eig}_4(K)$ only asks that the surface be isotopic to its own image. The authors build a Khovanov-type invariant from the Bar-Natan complex of a link together with its involution, using the Borel construction $\\partial_Q = \\partial + Q(1+\\tau)$. They prove that for the strongly invertible knot $J=17\\mathrm{nh}74$, $\\mathrm{eg}_4(\\#^m J) \\ge \\lceil m/2\\rceil$ while $\\mathrm{eig}_4(\\#^m J) \\le 1$ for every $m$. If correct, this is the first proof that the two genera differ, and that they differ by an arbitrarily large amount; it also explains why the standard mapping-cone invariants could not see the difference.","feed_headline":"Equivariant slice genus beats isotopy-equivariant genus by any amount","feed_subtitle":"A Borel-style Khovanov invariant sees true equivariant surfaces; earlier invariants could only see isotopic ones.","key_machinery":"The load-bearing object is the reduced Borel complex $\\widehat{\\mathrm{Kcr}}^-_Q(K) = (\\widehat{\\mathrm{Kcr}}^-(K) \\otimes \\mathbb{F}[Q],\\, \\partial_Q = \\partial + Q(1+\\tau))$, built from Bar-Natan homology of a transvergent diagram, that is, a symmetric projection whose symmetry axis is visible. The extra variable $Q$ makes the involution part of the differential, so the invariant remembers the higher homotopy-commutation behavior of $\\tau$ that ordinary $\\tau$-complexes forget. Its localization at $u$ is $\\mathbb{F}[u,u^{-1},Q]$, and the degrees in which $u$-nontorsion classes appear define the $\\mathrm{es}_{Q,A,B}$ invariants. Invariance is proved by an explicit analysis of equivariant Reidemeister moves M1, M2, and M3; locality of connected equivariant cobordism maps is what produces the genus bounds.","core_discovery":"This paper's central discovery is that the Borel complex $\\widehat{\\mathrm{Kc}}^-_Q(L) = (\\widehat{\\mathrm{Kc}}^-(L) \\otimes \\mathbb{F}[Q],\\, \\partial_Q = \\partial + Q(1+\\tau))$ is a well-defined invariant of an involutive link up to Sakuma equivalence, and that an equivariant cobordism $\\Sigma$ induces a local map on the reduced Borel complex $\\widehat{\\mathrm{Kcr}}^-_Q$ with grading shift $(0,-2g(\\Sigma))$. The localization condition $u^{-1}H_*(\\widehat{\\mathrm{Kcr}}^-_Q(K)) \\cong \\mathbb{F}[u,u^{-1},Q]$ yields refined numerical invariants $\\mathrm{es}_{Q,A,B}(K)$ satisfying the genus bound $\\mathrm{es}_{Q,A,B}(K_1) - 2g(\\Sigma) \\le \\mathrm{es}_{Q,A,B}(K_2)$. The main theorem is obtained by computing enough of the Borel complex of $J=17\\mathrm{nh}74$: it contains a fixed five-generator subcomplex $C_Q$, and the $m$-fold connected sum of that subcomplex has $\\mathrm{es}_{Q,m,m+1}(C_Q^{\\otimes m}) \\ge 2\\lceil m/2\\rceil$. The connected-sum formula is proved through Koszul duality, and a stabilization argument with the symmetric pair of slice disks for $J$ shows $\\mathrm{eig}_4(\\#^m J) \\le 1$.","pith_inferences":["Beyond the paper, the unbounded gap mechanism is portable: any knot whose reduced Borel complex contains a local image of the five-generator model $C_Q$ will produce the same growth under connected sums, so the phenomenon should occur in infinite families rather than in a single example.","Beyond the paper, the Koszul-duality connected-sum formula suggests that equivariant Khovanov connected sums are governed by the full $\\tau$-complexes of the factors, not their Borel complexes alone, which may create computational shortcuts for connected-sum computations.","Beyond the paper, a direct testable extension is to compute the Q-equivalence classes of mixed complexes for small strongly invertible knots, especially ones whose $\\tau$ action on Khovanov homology is trivial, to see whether the axis filtration alone distinguishes involutions that the Borel construction cannot yet distinguish."],"forward_implications":["For the knot $J=17\\mathrm{nh}74$, each connected sum $\\#^m J$ satisfies $\\mathrm{eg}_4(\\#^m J)\\ge\\lceil m/2\\rceil$ and $\\mathrm{eig}_4(\\#^m J)\\le1$, so the two symmetric slice genera differ and the gap is unbounded.","The invariants $\\mathrm{es}_{Q,A,B}(K)$ are equivariant concordance invariants and obey $\\mathrm{es}_{Q,A,B}(K_1)-2g(\\Sigma)\\le \\mathrm{es}_{Q,A,B}(K_2)$ for equivariant cobordisms, strictly refining the earlier mapping-cone invariants.","Because only genuine equivariant cobordisms induce Borel cobordism maps, the Borel invariants can distinguish the true equivariant slice genus from the isotopy-equivariant one; the mapping-cone style invariants cannot.","The mixed complex, whose Q-equivalence class is invariant and which carries local equivariant cobordism maps, records the Lobb-Watson axis filtration and gives a route to equivariant genus bounds for knots whose $\\tau$ action on homology is trivial.","An equivariantly squeezed knot must have $\\mathrm{es}(K)=s(K)$; in particular the strongly invertible knot $10_{141}$ is not equivariantly squeezed."],"supporting_citations":[{"why":"Supplies the transvergent diagram move framework, the involutive Reidemeister moves, and the axis filtration whose localizations the mixed complex remembers.","marker":"[L W21]"},{"why":"Constructs the mapping-cone Khovanov invariant for involutive links that the Borel complex refines, and supplies the tau-complex formalism and the basic es invariant.","marker":"[San25]"},{"why":"Introduces the isotopy-equivariant slice genus and invariants that only bound eig4, the comparison target of the main theorem.","marker":"[DMS23]"},{"why":"Provides Bar-Natan homology, the tangle category, and the explicit Reidemeister move maps used in the M1, M2, and M3 invariance arguments.","marker":"[BN05]"},{"why":"Gives the computer calculation of the tau action on Khovanov homology and the Bar-Natan spectral sequence used to constrain the Borel complex of specific knots.","marker":"[BO25]"},{"why":"Provides the classification of strong inversions and the eta-polynomial used to distinguish involutions and for the topological analogue of the genus gap.","marker":"[Sak86]"},{"why":"Supplies the Koszul duality equivalence used to prove the connected-sum formula for Borel complexes.","marker":"[BGS96]"},{"why":"Gives the splitting of the unreduced Bar-Natan complex into two reduced copies, enabling reduced Borel cobordism maps without path data.","marker":"[Wig16]"}],"fun_headline_variants":["Equivariant slice genus can exceed isotopy version arbitrarily","New Khovanov invariant reveals unbounded equivariant genus gap","Equivariant vs isotopy slice genus gap can be arbitrarily large","Khovanov refinement makes equivariant slice genus gap unbounded","Borel complex reveals unbounded gap between equivariant slice genera"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that every strongly invertible link and every equivariant cobordism can be equivariantly isotoped into the special symmetric diagram or movie form used here, with all the deferred genericity and move-level checks supplied rigorously; if any part of that fails, the Borel and mixed invariants are not known to be well-defined.","fun_headline_variants_meta":{"raw":{"variants":["Equivariant slice genus can exceed isotopy version arbitrarily","New Khovanov invariant reveals unbounded equivariant genus gap","Equivariant vs isotopy slice genus gap can be arbitrarily large","Khovanov refinement makes equivariant slice genus gap unbounded","Borel complex reveals unbounded gap between equivariant slice genera"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00056,"raw_usage":{"total_tokens":2653,"prompt_tokens":931,"completion_tokens":1722,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":547,"completion_tokens_details":{"reasoning_tokens":1636}},"tokens_in":547,"tokens_out":1722,"duration_ms":11840,"temperature":1.0,"reasoning_tokens":1636,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T16:19:41.406072+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the enumeration described in Lemma 6.13 on a transvergent diagram of $J=17\\mathrm{nh}74$ and check for a grading-shift-zero local map from the five-generator complex $C_Q$ into $\\widehat{\\mathrm{Kcr}}^-_Q(J)$; absence of such a map would break Lemma 6.13 and the bound $\\mathrm{eg}_4(\\#^mJ)\\ge\\lceil m/2\\rceil$.","supporting_citations":[],"review_version":1}