{"id":"67bd331d-3484-4325-9dc3-756ffd813bce","arxiv_id":"1908.05018","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The Kronheimer-Mrowka instanton invariant s# is not additive under connected sums and differs from Rasmussen's s, and new link invariants s#_+ and s#_- are constructed and computed for torus links.","lead":"This paper computes a gauge-theoretic knot invariant called s# for algebraic knots, torus links, and connected sums, and shows it behaves differently from the better-known s invariant. The author also splits s# into two new concordance invariants that satisfy a simple bound and generalize to links.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection to the central non-additivity claim; the reader's link-case worry affects only the paper's link computations.","rationale":"The reader's verdict of CONDITIONAL is reasonable because the paper explicitly relies on an unproved extension of the Kronheimer-Mrowka non-vanishing theorem from knots to links. The paper itself admits the gap in Section 5.1: 'In [3], Kronheimer and Mrowka only use this result for knots, but the proof applies to links as well.' This admission is exactly the kind of missing support the review should flag, and it directly affects the paper's computations for torus links and the new link invariants. However, the reader's weakest assumption states that this gap undermines the central algebraic-knot computation and the non-additivity conclusion. That part is not accurate: the algebraic knots and the connected-sum knots used for non-additivity have a single boundary component, so the knot case of the theorem suffices. For knots, the normalization used in Lemma 5.1 can be recovered from the embedded curve's non-vanishing and the equality of the two composite immersed cobordisms, without needing a separate non-vanishing theorem for the immersed curve. Thus the central claim appears sound. The paper's overall verdict should remain CONDITIONAL because the unproved link extension still affects substantial stated results, but the concern is narrower than the reader's weakest assumption suggests.","tokens_in":19163,"tokens_out":36593,"duration_ms":350532,"concrete_test":"Verify the central claim by re-deriving Lemma 5.1 for the knot case without invoking the link extension: use the equality ψ(T)∘ψ(Σ0_1)=ψ(S)∘ψ(Σ0_2) together with the [3, §5.4] non-vanishing for the embedded Σ0_2 and integrality over Q[[λ]] to solve for the valuations of ψ(Σ0_1); if the solution forces m+(Σ0_1)=0 and m-(Σ0_1)=1, the non-additivity computation stands. Separately, check the link extension on the Hopf link by computing whether the map induced by its punctured singular complex curve has at least one m±=0; if not, the link-level theorems need revision.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Lemma 5.1 the paper states that the non-vanishing theorem of [3, §5.4] applies to the punctured complex curves Σ0_1 and Σ0_2, adding 'In [3], Kronheimer and Mrowka only use this result for knots, but the proof applies to links as well.' This is an explicit, unproved extension. However, the paper's central claim — non-additivity of s# on knots — does not rest on it. The relevant knots T_{p,q}#T_{p,q} have connected boundary, so only the knot case of the theorem is needed. Moreover, for a knot the normalization m+(Σ0_1)=0 and m-(Σ0_1)=1 follows from the embedded curve's non-vanishing together with the equality of the two composite cobordisms (T∘Σ0_1 and S∘Σ0_2) and the fact that the comparison is over Q[[λ]]; no separate non-vanishing for the immersed curve is required. Thus the reader's flagged gap is real but is not load-bearing for the headline result. It does affect the link invariants and the computations for torus links (Corollary 5.4, Theorem 1.9), so the paper as a whole still merits a conditional verdict pending a proof of the link case.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper defines link generalizations of Kronheimer and Mrowka's instanton concordance invariant s^#, namely the 2^l invariants s^#_I and the pair s^#_+, s^#_-, using counts of lambda-divisibility of maps induced by immersed cobordisms. It proves cobordism inequalities, bounds on s^#_+ - s^#_-, and computations for algebraic knots, connected sums of quasi-positive knots with right-handed torus knots, and torus links. The headline result is that s^# is not additive: for a right-handed torus knot T_{p,q}, Corollary 5.6 gives s^#(T_{p,q}#T_{p,q}) = 4g-1 while 2s^#(T_{p,q}) = 4g-2. The paper also proposes s^#_+ - s^#_- as a new link invariant and gives a characterization in terms of the map induced by a genus-one cobordism.","tokens_in":19349,"tokens_out":8874,"duration_ms":89970,"significance":"If the computations are correct, the non-additivity of s^# is a substantial and surprising result, since it distinguishes s^# from Rasmussen's concordance homomorphism s. The new link invariants and the difference s^#_+ - s^#_- are natural and potentially useful concordance data, and the paper gives explicit, concrete formulas for them on torus links. The paper is organized carefully and builds on established theorems rather than on its own conclusions; I see no circularity. The main computations are supported by detailed constructions, but the proof relies on an explicitly unproved extension of a non-vanishing theorem from knots to links, and one key computation in Theorem 6.1 is only sketched. These issues do not undermine the headline non-additivity claim, but they do affect the link-theoretic results.","major_comments":[{"comment":"The proof invokes the non-vanishing result of [3, §5.4] to assert that both psi^#(Sigma^0_1) and psi^#(Sigma^0_2) are not divisible by lambda, adding that 'In [3], Kronheimer and Mrowka only use this result for knots, but the proof applies to links as well.' This is an unproved extension of a nontrivial theorem, and it is load-bearing for the link computations in Corollary 5.4 and Theorem 6.1. Moreover, Sigma^0_1 is an immersed surface with a transverse double point, so the statement being invoked must also cover immersed cobordisms; the manuscript does not identify the precise statement from [3] or explain why its proof covers this case. The main non-additivity result for T_{p,q}#T_{p,q} uses only the knot case of Lemma 5.1, so I do not view this gap as fatal to the headline claim, but the link-theoretic conclusions are not yet justified.","section":"Section 5.1, Lemma 5.1"},{"comment":"The final paragraph of the proof asserts that 'by comparing Sigma_1 ∘ D_1 and Sigma ∘ T_1, as in Lemma 5.1, we can deduce that m_I(Sigma) is the number of instances of u_- that appears in u_I.' This is the core of the computation of s^#_I for torus links, but the deduction is not shown. In particular, the observation that some generator maps to a non-multiple of lambda does not by itself imply that the all-u_+ generator maps to a non-multiple, and the displayed comparison must be worked out for each index I. The proof also appears to reuse the unproved link-case non-vanishing statement. This step needs to be expanded before Theorem 1.9 can be verified.","section":"Section 6, Theorem 6.1"}],"minor_comments":[{"comment":"There are several typographical slips: 'the invariants♯ defines a map' should read 'the invariant s^# defines a map', and 'the invariants♯ for links' should be 'the invariant s^# for links' or similar.","section":"Introduction and abstract"},{"comment":"The cobordism C and C' are defined for a knot K, but the displayed matrices use s^#_+(L)-s^#_-(L); the symbol L should be replaced by K for consistency.","section":"Section 3.2, Proposition 3.5"},{"comment":"The proof uses v_+ and v_- for a general link L, but these generators were defined only for knots in Section 2; a short definition for the relevant component of a link would remove ambiguity.","section":"Section 3.2, proof of Proposition 3.4"},{"comment":"The sentence 'Note that the double point had to be a positive double point' is asserted without explanation; a brief justification using the complex orientation or the sign convention in [4] would be helpful.","section":"Section 5.1, Lemma 5.1"},{"comment":"The statement says every component of Sigma has non-trivial boundary in L1, while the proof assumes the components have boundary in both L1 and L2; these hypotheses should be reconciled explicitly.","section":"Section 4.2, Theorem 1.4"}],"recommendation":"major_revision","confidential_remarks":"The main non-additivity result appears sound and is a strong contribution, but the paper should be sent back for revision so that the author can either prove or properly cite the link-case non-vanishing theorem and expand the final step of Theorem 6.1. If those gaps are filled, the paper would be suitable for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nRead Gong's paper. If you work on concordance invariants, it is worth your time. The headline result is that s# is not additive under connected sum: for a right-handed torus knot T_{p,q}, s#(T_{p,q}#T_{p,q}) = 4g−1 while 2s#(T_{p,q}) = 4g−2. That is a clean, surprising contrast with Rasmussen's s, and the argument for it holds up as far as I can tell.\n\nThe paper's real novelty is splitting s# into s#+ and s#−, proving that s#+ − s#− lies in {0,1,2}, and using immersed cobordisms to get link invariants s#± and s#_I. The computations for algebraic knots, connected sums of quasi-positive knots with torus knots, and torus links are genuine and substantial. The proof of Proposition 3.4 (the 0–2 bound) is a nice piece of bookkeeping, and the use of singular complex curves is appropriate and well-motivated. There is no circularity: the invariants are built from established instanton homology, and the paper cites the relevant literature fairly.\n\nThe soft spot is the one the reader caught. In Lemma 5.1, the paper says the Kronheimer–Mrowka nonvanishing theorem applies to links as well as knots, adding that the proof applies to links, but no proof or citation is given. That is a real gap. However, the stress-test note is right: the central non-additivity claim does not rest on it, because T_{p,q}#T_{p,q} has connected boundary and only the knot case is needed. The link extension is load-bearing for the torus-link computations (Corollary 5.4, Theorem 1.9) and for the s#_I values, so those parts are conditional as the reader says. Also, Proposition 5.8 goes a bit quickly: the step from the composite map having no λ factors to the individual maps having no λ factors is valid but deserves a sentence.\n\nThe citation pattern is clean. The paper is honest about what is open (e.g., additivity of s#±). The proofs are mostly detailed, and the few terse steps, like the end of Theorem 6.1, are recoverable.\n\nVerdict: this deserves peer review, not desk rejection. I would send it to a competent referee and ask the author to supply the missing link-case nonvanishing statement, either by proof or by a precise citation. With that fixed, the paper is solid. Even without fixing the link extension, the knot non-additivity result stands and is itself a significant contribution.\n\nBring it to reading group.","headline":"A serious, mostly convincing paper: the non-additivity of s# is real and the new s#± invariants are worth knowing, but the link-case extension of Kronheimer–Mrowka nonvanishing is an explicit gap that should be patched before publication.","tokens_in":19931,"tokens_out":2086,"would_cite":true,"duration_ms":23074,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["57K10","57R58"],"pacs":[],"model":"deepseek-v4-flash","headline":"The gauge-theoretic concordance invariant $s^\\sharp$ is not additive under connected sums: for a right-handed torus knot of genus $g$, $s^\\sharp(T_{p,q}\\#T_{p,q})=4g-1$ while $2s^\\sharp(T_{p,q})=4g-2$.","keywords":["s sharp invariant","singular instanton homology","concordance group","slice genus","torus knots","algebraic knots","link concordance invariants","immersed cobordisms"],"falsifier":"Compute $s^\\sharp$ on the connected sum of two right-handed trefoils by an independent route — for example, a direct count of $\\lambda$ factors in the map induced by a genus-two cobordism from the unknot to $T_{2,3}\\#T_{2,3}$, or an exact-triangle calculation. The paper's claim forces the answer to be $3$; additivity would force the answer to be $2$, and the two predictions are not compatible.","tokens_in":18903,"feed_emoji":"⚓","tokens_out":12714,"duration_ms":106879,"temperature":0.7,"pith_summary":"The paper computes the gauge-theoretic knot invariant $s^\\sharp$ on algebraic knots and on connected sums of quasi-positive knots with non-trivial right-handed torus knots, and derives a consequence: $s^\\sharp$ is not additive under connected sum. For a right-handed torus knot $T_{p,q}$ of genus $g$, the computation gives $s^\\sharp(T_{p,q}\\#T_{p,q})=4g-1$, while $2s^\\sharp(T_{p,q})=4g-2$. This matters because $s^\\sharp$ was introduced as the instanton-theoretic analogue of Rasmussen's $s$, and $s$ is a homomorphism from the concordance group to $\\mathbb{Z}$; the paper shows that a concordance invariant which still lower-bounds the slice genus can fail to be a homomorphism. Along the way the author splits $s^\\sharp$ into two invariants $s^\\sharp_+$ and $s^\\sharp_-$ whose difference lies in $\\{0,1,2\\}$, and extends both to links using immersed cobordisms.","feed_headline":"Knot invariant s# breaks connected-sum additivity","feed_subtitle":"For T#T the value is 4g−1, not 4g−2, so the gauge-theoretic concordance invariant cannot be a homomorphism.","key_machinery":"The mechanism is the $\\lambda$-adic valuation of cobordism-induced maps on the torsion-free part $I'(L)$ of singular instanton homology with local coefficients, a free module over $\\mathbb{Q}[[\\lambda]]$ with $2^l$ generators for an $l$-component link. For an immersed cobordism $\\Sigma$ from an unlink to $L$, $s^\\sharp_I(L)$ is defined as the genus of $\\Sigma$ plus its number of positive double points minus the maximum power of $\\lambda$ dividing the image of the generator $u_I$; $s^\\sharp_\\pm$ is the same count for the two generators of the unknot, with a parity correction. The local-move rule of Kronheimer and Mrowka — a positive twist multiplies the induced map by $1-u^2$, a negative twist does not change it, and a finger move multiplies by $1-u^2$ — is what makes the count independent of the chosen cobordism. Lemma 5.1 compares the maps induced by a singular complex curve and its embedded perturbation and uses the non-vanishing theorem to force the two generators to pick up exactly zero and one powers of $\\lambda$, which is the arithmetic fact behind $s^\\sharp(L)=2g-1$.","core_discovery":"On its own terms, the central discovery is that the map $s^\\sharp$ from smooth concordance classes of knots to $\\mathbb{Z}$ is not a homomorphism: for $T_{p,q}\\#T_{p,q}$ the value is $4g-1$, not $4g-2$. The route to this is a computation for singular complex curves. Lemma 5.1 proves that if a link $L$ bounds an immersed complex curve in $B^4$ that is embedded except for one positive transverse double point and is irreducible in the ball, then $s^\\sharp_+(L)=g$ and $s^\\sharp_-(L)=g-1$, so $s^\\sharp(L)=2g-1$. Algebraic knots, the relevant connected sums, and many torus links are shown to bound such curves, yielding the formula. The paper also introduces the two link invariants $s^\\sharp_I$ and $s^\\sharp_\\pm$, proves the cobordism inequalities of Theorems 1.2 and 1.4, and computes $s^\\sharp_I$ for torus links $T_{md,nd}$ explicitly.","pith_inferences":["Because $s^\\sharp$ is modelled on Rasmussen's $s$, its failure of additivity suggests that gauge-theoretic concordance invariants can see structure in the concordance group that concordance homomorphisms average out; a natural next test, left open by the paper, is whether $s^\\sharp$ is an almost homomorphism with a universal defect bound.","The difference $s^\\sharp_+-s^\\sharp_-$ could serve as a concordance invariant that separates knots with equal slice genus; checking whether it vanishes on all slice knots or detects chirality would be a direct extension.","The immersed-cobordism formulation suggests a computational strategy for other gauge-theoretic invariants: replace embedded surfaces by immersed ones with positive double points, count $\\lambda$-factors, and use the local-move rule to control the change.","The link invariants $s^\\sharp_I$ may yield component-wise concordance obstructions, since Theorem 1.2 depends on the parity of the genus on each component and on which of the $2^l$ generators is tracked."],"forward_implications":["For a right-handed torus knot $T_{p,q}$, $s^\\sharp(T_{p,q})=2g-1$ while Rasmussen's $s(T_{p,q})=2g$, so the two invariants genuinely disagree even though both lower-bound the slice genus.","$s^\\sharp$ is a concordance invariant that is not a group homomorphism, so the concordance group admits a slice-genus bound of this type that cannot be factored through a homomorphism to $\\mathbb{Z}$.","The split invariants satisfy $0\\le s^\\sharp_+(L)-s^\\sharp_-(L)\\le 2$ for every link, and the difference is a link invariant taking only three values.","Theorem 1.4 gives a lower bound on the genus plus positive double points of any immersed cobordism between two links; for embedded cobordisms it reduces to a genus bound in the spirit of the slice-genus bound.","For torus links $T_{md,nd}$, the $2^l$ invariants $s^\\sharp_I$ are completely determined by $m,n,d$ and the signs in $I$, with an explicit parity switch."],"supporting_citations":[{"why":"Introduces $s^\\sharp$ and the instanton homology $I^\\sharp(L;\\Gamma)$ with local coefficients, together with the local-move formula for cobordism maps that the paper uses throughout.","marker":"[4]"},{"why":"Provides the non-vanishing theorem for maps induced by punctured complex curves; Lemma 5.1 extends it from knots to links and leans on it for the $\\lambda$-valuation counts.","marker":"[3]"},{"why":"Rudolph's construction of quasi-positive knots as boundaries of algebraic curves is used in Lemma 5.5 to build the singular complex curve bounding $K\\#T_{p,q}$.","marker":"[7]"},{"why":"Defines Rasmussen's $s$, the homomorphism from the concordance group to $\\mathbb{Z}$ that $s^\\sharp$ is compared with and shown not to imitate.","marker":"[6]"}],"fun_headline_variants":["s# not additive under connected sums, explicit counterexample","Kronheimer-Mrowka s# breaks homomorphism for knots","T#T shows s# non-additive; new invariants split it","s# non-additive: T#T gives 4g-1 not 4g-2"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the non-vanishing theorem for punctured complex curves, proved in the cited literature for knots, extends to links; Lemma 5.1 uses that extension to conclude that not every generator of the link's instanton homology is divisible by $\\lambda$, and without it the formula $s^\\sharp(L)=2g-1$ for algebraic knots and the non-additivity of $s^\\sharp$ would lose their foundation.","fun_headline_variants_meta":{"raw":{"variants":["s# not additive under connected sums, explicit counterexample","Kronheimer-Mrowka s# breaks homomorphism for knots","T#T shows s# non-additive; new invariants split it","s# non-additive: T#T gives 4g-1 not 4g-2"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000787,"raw_usage":{"total_tokens":3544,"prompt_tokens":1088,"completion_tokens":2456,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":704,"completion_tokens_details":{"reasoning_tokens":2373}},"tokens_in":704,"tokens_out":2456,"duration_ms":20255,"temperature":1.0,"reasoning_tokens":2373,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:25:20.199587+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute $s^\\sharp$ on the connected sum of two right-handed trefoils by an independent route — for example, a direct count of $\\lambda$ factors in the map induced by a genus-two cobordism from the unknot to $T_{2,3}\\#T_{2,3}$, or an exact-triangle calculation. The paper's claim forces the answer to be $3$; additivity would force the answer to be $2$, and the two predictions are not compatible.","supporting_citations":[{"cited_title":"Beliakova and S","cited_arxiv_id":null,"evidence_quote":"Introduces $s^\\sharp$ and the instanton homology $I^\\sharp(L;\\Gamma)$ with local coefficients, together with the local-move formula for cobordism maps that the paper uses throughout."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the non-vanishing theorem for maps induced by punctured complex curves; Lemma 5.1 extends it from knots to links and leans on it for the $\\lambda$-valuation counts."},{"cited_title":"Gauge theory and Rasmussen's invariant","cited_arxiv_id":"1110.1297","evidence_quote":"Rudolph's construction of quasi-positive knots as boundaries of algebraic curves is used in Lemma 5.5 to build the singular complex curve bounding $K\\#T_{p,q}$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines Rasmussen's $s$, the homomorphism from the concordance group to $\\mathbb{Z}$ that $s^\\sharp$ is compared with and shown not to imitate."}],"review_version":1}