On the Kronheimer-Mrowka concordance invariant

Sherry Gong

REVIEW 2 major objections 5 minor 10 references

On the Kronheimer-Mrowka concordance invariant

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read 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 $2s^\sharp(T_{p,q})=4g-2$.

desk verdict 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. read the letter →

arxiv 1908.05018 v1 pith:BOGXATXI submitted 2019-08-14 math.GT

Sherry Gong This is my paper · ORCID

classification math.GT MSC 57K1057R58

keywords ssharpinvariantsingularinstantonhomologyconcordancegroupslicegenustorusknotsalgebraiclinkinvariantsimmersedcobordisms

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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 $s^\sharp(L)=2g-1$.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

Read 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.

The 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.

The 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.

The 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.

Verdict: 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.

Bring it to reading group.

Referee Report

2 major / 5 minor

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.

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 (2)

[Section 5.1, Lemma 5.1] 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 6, Theorem 6.1] 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.

minor comments (5)

[Introduction and abstract] 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 3.2, Proposition 3.5] 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 3.2, proof of Proposition 3.4] 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 5.1, Lemma 5.1] 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 4.2, Theorem 1.4] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: computations rest on external instanton-homology theorems and complex-curve constructions; the link-case gap is a rigor issue, not circularity.

full rationale

The derivation chain is self-contained with respect to external inputs: s# and the induced maps are imported from Kronheimer-Mrowka [4], and the non-vanishing input in Lemma 5.1 is cited to [3, Section 5.4]. The central non-additivity claim does not use that cited result in a circular way: Corollary 5.6 combines the external computation s#(T_{p,q}) = 2g - 1 for a right-handed torus knot (as an algebraic knot) with the new computation s#(K # T_{p,q}) = 2g - 1 for quasi-positive K, yielding s#(T_{p,q} # T_{p,q}) = 4g - 1, while 2s#(T_{p,q}) = 4g - 2. No fitted parameter or definitional identity forces this discrepancy. The paper's own definitions of s#_+, s#_-, and s#_I are built from the same Kronheimer-Mrowka maps, and the inequalities and computations are proved by comparing induced maps, not by assuming the target values. The only substantive caveat is Lemma 5.1's sentence: 'In [3], Kronheimer and Mrowka only use this result for knots, but the proof applies to links as well.' That is an unproved extension of an external theorem; it affects link computations such as Corollary 5.4 and Theorem 1.9, but it is a correctness or rigor concern, not a circularity concern, and it does not bear on the knot-based non-additivity conclusion. There is no load-bearing self-citation, no renaming of a known result as a new invariant, and no fitted-input-called-prediction pattern. The manuscript is not circular.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters were fitted; all numerical choices in the paper are generic perturbation parameters rather than tuned values. The central claims rest on standard theorems in instanton homology and complex geometry from the cited literature, plus one unproved extension, the link case of the non-vanishing theorem. The new invariants s#_+ and s#_- are mathematical definitions, not postulated physical entities.

assumptions (5)

domain assumption The non-vanishing result of Section 5.4 of [3] applies to links, not only to knots.
Invoked in the proof of Lemma 5.1 to conclude that the maps induced by punctured complex curves are not divisible by lambda. The paper states that the proof carries over to links but does not provide the details, making it a load-bearing assumption.
standard math The excision/Kunneth splitting I'(L1 disjoint union L2) is isomorphic to I'(L1) tensor I'(L2), from [2].
Used in Section 3.1 to identify the generators of I'(U_l) and in Proposition 3.4 to prove the bound on s#_+ - s#_-.
standard math The local moves on immersed cobordisms change the induced maps as described in Proposition 3.1 of [4].
Used throughout Section 3 to prove well-definedness of s#_I and s#_+/- and to compare maps from different cobordisms.
standard math The genus-one cobordism T induces psi#(T)(u_+) = 2u_- and psi#(T)(u_-) = 2lambda^2 u_+, as computed in [4].
Used in Lemma 5.1 and Proposition 3.5 to compute s# and the difference s#_+ - s#_-.
standard math Rudolph's theorem provides a complex curve bounding any quasi-positive knot K, as in [7].
Used in Lemma 5.5 to construct a singular complex curve bounding K#T_{p,q} for a quasi-positive knot K.

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Cite this review

Pith. "Pith review of On the Kronheimer-Mrowka concordance invariant." pith.science (2026). https://pith.science/paper/BOGXATXI

@misc{pith2026190805018,
  author       = {Pith},
  title        = {Pith review of: On the Kronheimer-Mrowka concordance invariant},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BOGXATXI}},
  note         = {Machine review of arXiv:1908.05018}
}

abstract

Kronheimer and Mrowka introduced a new knot invariant, called $s^\sharp$, which is a gauge theoretic analogue of Rasmussen's $s$ invariant. In this article, we compute Kronheimer and Mrowka's invariant for some classes of knots, including algebraic knots and the connected sums of quasi-positive knots with non-trivial right handed torus knots. These computations reveal some unexpected phenomena: we show that $s^\sharp$ does not have to agree with $s$, and that $s^\sharp$ is not additive under connected sums of knots. Inspired by our computations, we separate the invariant $s^\sharp$ into two new invariants for a knot $K$, $s^\sharp_+(K)$ and $s^\sharp_-(K)$, whose sum is $s^\sharp(K)$. We show that their difference satisfies $0 \leq s^\sharp_+(K) - s^\sharp_-(K) \leq 2$. This difference may be of independent interest. We also construct two link concordance invariants that generalize $s^\sharp_\pm$, one of which we continue to call $s^\sharp_\pm$, and the other of which we call $s^\sharp_I$. To construct these generalizations, we give a new characterization of $s^\sharp$ using immersed cobordisms rather than embedded cobordisms. We prove some inequalities relating the genus of a cobordism between two links and the invariant $s^\sharp$ of the links. Finally, we compute $s^\sharp_\pm$ and $s^\sharp_I$ for torus links.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

10 extracted references · 8 canonical work pages

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [3]

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@esa (Ref

\@ifxundefined[1] #1\@undefined \@firstoftwo \@secondoftwo \@ifnum[1] #1 \@firstoftwo \@secondoftwo \@ifx[1] #1 \@firstoftwo \@secondoftwo [2] @ #1 \@temptokena #2 #1 @ \@temptokena \@ifclassloaded agu2001 natbib The agu2001 class already includes natbib coding, so you should not add it explicitly Type <Return> for now, but then later remove the command n...

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[4] [4]

Beliakova and S

A. Beliakova and S. Wehrli. Categorification of the colored J ones polynomial and R asmussen invariant of links. Canad. J. Math. , 60(6):1240--1266, 2008

work page 2008

[5] [5]

P. B. Kronheimer and T. S. Mrowka. Khovanov homology is an unknot-detector. Publ. Math. Inst. Hautes \' E tudes Sci. , (113):97--208, 2011

work page 2011

[6] [6]

P. B. Kronheimer and T. S. Mrowka. Knot homology groups from instantons. J. Topol. , 4(4):835--918, 2011

work page 2011

[7] [7]

P. B. Kronheimer and T. S. Mrowka. Gauge theory and R asmussen's invariant. J. Topol. , 6(3):659--674, 2013. Corrigendum at arXiv:1110.1297v3

work page Pith review arXiv 2013

[8] [8]

J. Pardon. The link concordance invariant from L ee homology. Algebr. Geom. Topol. , 12(2):1081--1098, 2012

work page 2012