Bennequin-Plamenevskaya-Shumakovitch type inequalities for Kronheimer-Mrowka's concordance invariant

Nobuo Iida

arxiv: 2309.00415 · v3 · submitted 2023-09-01 · 🧮 math.GT

Bennequin-Plamenevskaya-Shumakovitch type inequalities for Kronheimer-Mrowka's concordance invariant

Nobuo Iida This is my paper

Pith reviewed 2026-05-24 07:03 UTC · model grok-4.3

classification 🧮 math.GT

keywords concordance invariantsslice-torus invariantsBennequin inequalitytorus knotss# invariantcobordism inequality

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The pith

Kronheimer-Mrowka concordance invariant s# obeys Bennequin-Plamenevskaya-Shumakovitch lower bounds.

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

The paper establishes lower bounds for the concordance invariant s# that mirror the classical Bennequin-Plamenevskaya-Shumakovitch inequalities. These bounds follow from computing s# on torus knots and applying the cobordism inequality for s# proved by Gong. The proof uses the same arguments previously applied to other slice-torus invariants. A sympathetic reader would care because the bounds supply diagram-based estimates for s# that can obstruct sliceness and bound concordance distance.

Core claim

The concordance invariant s# satisfies inequalities of Bennequin-Plamenevskaya-Shumakovitch type. This follows directly from its explicit values on torus knots together with the cobordism inequality due to Gong, using the standard transfer arguments for slice-torus invariants.

What carries the argument

Gong's cobordism inequality for s#, which transfers the known values on torus knots to general knots via suitable cobordisms.

If this is right

For any knot K, s#(K) is bounded below by an expression depending on the writhe and Seifert circles of any diagram of K.
The bounds give obstructions to a knot being slice.
The inequalities take exactly the same form as the classical ones on torus knots.
Any knot cobordant to a torus knot inherits the corresponding lower bound.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same method may produce diagram bounds for other concordance invariants in the Kronheimer-Mrowka family.
If the bounds are sharp on some infinite families, they could determine the exact value of s# from diagrams alone.
These bounds allow s# to replace older slice-torus invariants in any argument that relies only on diagram estimates.

Load-bearing premise

Gong's cobordism inequality for s# holds for the cobordisms that arise in the standard slice-torus arguments, with no additional obstructions specific to s#.

What would settle it

A concrete knot for which a Bennequin-Plamenevskaya-Shumakovitch expression computed from a diagram exceeds the actual value of s# on that knot.

read the original abstract

We give Bennequin-Plamenevskaya-Shumakovich type lower bounds for the concordance invariant $s^{\#}$ introduced by Kronheimer and Mrowka. The proof is a consequence of computations for torus knots and the cobordism inequality of $s^{\#}$ due to Gong, combined with well-known arguments used for slice-torus invariants.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This short note derives explicit lower bounds for s# by combining torus-knot data with Gong's cobordism inequality and standard slice-torus arguments.

read the letter

The paper's main contribution is a set of Bennequin-Plamenevskaya-Shumakovitch style lower bounds on the s# concordance invariant. It obtains them by feeding known torus-knot values into Gong's cobordism inequality and running the usual slice-torus transfer arguments. That combination produces new explicit statements that were not already in the literature, even though the proof steps themselves are recycled from earlier work on slice-torus invariants. The write-up is short and direct, which matches the modest scope. The argument structure looks internally consistent and does not introduce new technical machinery or hidden fitting steps. The only real dependence is on the external torus-knot computations and on Gong's inequality applying to the relevant cobordisms; the note treats both as given, which is reasonable for a consequence paper. No circularity or unstated assumption appears to break the chain. Readers who already work with s# or with slice-genus bounds will find the inequalities immediately usable for concrete calculations. People outside that narrow corner of knot theory will not need it. The paper is a clean, small addition rather than a major advance, but it is honest about what it does and what it relies on. I would send it to peer review; a referee can check the transfer details in a few pages and confirm the bounds are correctly stated.

Referee Report

0 major / 1 minor

Summary. The manuscript claims to establish Bennequin-Plamenevskaya-Shumakovitch type lower bounds for the concordance invariant s# introduced by Kronheimer and Mrowka. The argument is presented as a direct consequence of existing torus-knot computations, the cobordism inequality for s# proved by Gong, and the standard slice-torus arguments previously applied to other invariants; no new technical machinery is introduced.

Significance. If the result holds, the paper supplies explicit lower bounds on s# that parallel classical BPS inequalities, thereby extending the toolkit available for studying smooth concordance. The manuscript correctly credits the reliance on prior torus-knot data and Gong's independently established cobordism inequality; this reuse of verified external results is a strength for a short consequence note.

minor comments (1)

The abstract and title use slightly inconsistent transliterations of the name (Shumakovitch vs. Shumakovich); standardize the spelling throughout.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment, the recognition that the result follows from existing torus-knot data and Gong's cobordism inequality, and the recommendation to accept the manuscript.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The manuscript is a short note deriving BPS-type bounds for s# explicitly as a consequence of external torus-knot computations, Gong's independently established cobordism inequality, and previously published standard arguments for slice-torus invariants. No internal equations, fitted parameters, or self-citations are used to define or predict the claimed lower bounds; the derivation chain therefore remains self-contained against external benchmarks and does not reduce any result to quantities constructed inside the paper itself.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on two external results whose validity is assumed rather than re-derived: exact values of s# on torus knots and Gong's cobordism inequality. No free parameters or new entities are introduced in the abstract.

axioms (2)

domain assumption s# has been computed exactly on all torus knots.
The proof begins from these computations.
domain assumption Gong's cobordism inequality for s# holds.
Cited as an external input.

pith-pipeline@v0.9.0 · 5580 in / 1300 out tokens · 22625 ms · 2026-05-24T07:03:25.642631+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The proof is a consequence of computations for torus knots and the cobordism inequality of s# due to Gong, combined with well-known arguments used for slice-torus invariants.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Gong’s cobordism inequality … s#(L2) − s#(L1) ≤ −χ(Σ) + |L1| − |L2|

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

[1]

Tetsuya Abe, The Rasmussen invariant of a homogeneous knot , Proc. Amer. Math. Soc. 139 (2011), no. 7, 2647–2656. MR2784833

work page 2011
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Baldwin and Steven Sivek, Framed instanton homology and concordance , J

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Aliakbar Daemi, Hayato Imori, Kouki Sato, Christopher S caduto, and Masaki Taniguchi, Instantons, special cycles, and knot concordance , 2022

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Etnyre, Legendrian and transversal knots , Handbook of knot theory, 2005, pp

John B. Etnyre, Legendrian and transversal knots , Handbook of knot theory, 2005, pp. 105–

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Etnyre and Ko Honda, Knots and contact geometry

John B. Etnyre and Ko Honda, Knots and contact geometry. I. Torus knots and the ﬁgure eight knot , J. Symplectic Geom. 1 (2001), no. 1, 63–120. MR1959579

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[6]

Sherry Gong, On the Kronheimer-Mrowka concordance invariant , J. Topol. 14 (2021), no. 1, 1–28. MR4186131

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1, 29–38

Tomomi Kawamura, The Rasmussen invariants and the sharper slice-Bennequin i nequality on knots , Topology 46 (2007), no. 1, 29–38. MR2288725

work page 2007
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196 (2015), 558–574

, An estimate of the Rasmussen invariant for links and the dete rmination for certain links, Topology Appl. 196 (2015), 558–574. MR3430998

work page 2015
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P. B. Kronheimer and T. S. Mrowka, Gauge theory for embedded surfaces. I , Topology 32 (1993), no. 4, 773–826. MR1241873

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, Gauge theory for embedded surfaces. II , Topology 34 (1995), no. 1, 37–97. MR1308489

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, Khovanov homology is an unknot-detector , Publ. Math. Inst. Hautes ´Etudes Sci. 113 (2011), 97–208. MR2805599

work page 2011
[12]

, Knot homology groups from instantons , J. Topol. 4 (2011), no. 4, 835–918. MR2860345

work page 2011
[13]

, Gauge theory and Rasmussen ’s invariant , J. Topol. 6 (2013), no. 3, 659–674. MR3100886

work page 2013
[14]

Lukas Lewark, Khovanov-Rozansky homologies, knotted weighted webs and t he slice genus , PhD thesis

work page
[15]

, Rasmussen ’s spectral sequences and the slN -concordance invariants, Adv. Math. 260 (2014), 59–83. MR3209349

work page 2014
[16]

Charles Livingston, Computations of the Ozsv´ ath-Szab´ o knot concordance invariant, Geom. Topol. 8 (2004), 735–742. MR2057779

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[17]

Andrew Lobb, Computable bounds for Rasmussen ’s concordance invariant , Compos. Math. 147 (2011), no. 2, 661–668. MR2776617

work page 2011
[18]

Olga Plamenevskaya, Transverse knots and Khovanov homology , Math. Res. Lett. 13 (2006), no. 4, 571–586. MR2250492

work page 2006
[19]

Jacob Rasmussen, Khovanov homology and the slice genus , Invent. Math. 182 (2010), no. 2, 419–447. MR2729272

work page 2010
[20]

Lee Rudolph, Quasipositivity as an obstruction to sliceness , Bull. Amer. Math. Soc. (N.S.) 29 (1993), no. 1, 51–59. MR1193540

work page 1993
[21]

Shumakovitch, Rasmussen invariant, slice-Bennequin inequality, and sli ceness of knots , J

Alexander N. Shumakovitch, Rasmussen invariant, slice-Bennequin inequality, and sli ceness of knots , J. Knot Theory Ramiﬁcations 16 (2007), no. 10, 1403–1412. MR2384833 Tokyo Institute of Technology, Ookayama, Meguro-ku, Tokyo Email address : iida.n.ad@m.titech.ac.jp

work page 2007

[1] [1]

Tetsuya Abe, The Rasmussen invariant of a homogeneous knot , Proc. Amer. Math. Soc. 139 (2011), no. 7, 2647–2656. MR2784833

work page 2011

[2] [2]

Baldwin and Steven Sivek, Framed instanton homology and concordance , J

John A. Baldwin and Steven Sivek, Framed instanton homology and concordance , J. Topol. 14 (2021), no. 4, 1113–1175. MR4332488

work page 2021

[3] [3]

Aliakbar Daemi, Hayato Imori, Kouki Sato, Christopher S caduto, and Masaki Taniguchi, Instantons, special cycles, and knot concordance , 2022

work page 2022

[4] [4]

Etnyre, Legendrian and transversal knots , Handbook of knot theory, 2005, pp

John B. Etnyre, Legendrian and transversal knots , Handbook of knot theory, 2005, pp. 105–

work page 2005

[5] [5]

Etnyre and Ko Honda, Knots and contact geometry

John B. Etnyre and Ko Honda, Knots and contact geometry. I. Torus knots and the ﬁgure eight knot , J. Symplectic Geom. 1 (2001), no. 1, 63–120. MR1959579

work page 2001

[6] [6]

Sherry Gong, On the Kronheimer-Mrowka concordance invariant , J. Topol. 14 (2021), no. 1, 1–28. MR4186131

work page 2021

[7] [7]

1, 29–38

Tomomi Kawamura, The Rasmussen invariants and the sharper slice-Bennequin i nequality on knots , Topology 46 (2007), no. 1, 29–38. MR2288725

work page 2007

[8] [8]

196 (2015), 558–574

, An estimate of the Rasmussen invariant for links and the dete rmination for certain links, Topology Appl. 196 (2015), 558–574. MR3430998

work page 2015

[9] [9]

P. B. Kronheimer and T. S. Mrowka, Gauge theory for embedded surfaces. I , Topology 32 (1993), no. 4, 773–826. MR1241873

work page 1993

[10] [10]

II , Topology 34 (1995), no

, Gauge theory for embedded surfaces. II , Topology 34 (1995), no. 1, 37–97. MR1308489

work page 1995

[11] [11]

, Khovanov homology is an unknot-detector , Publ. Math. Inst. Hautes ´Etudes Sci. 113 (2011), 97–208. MR2805599

work page 2011

[12] [12]

, Knot homology groups from instantons , J. Topol. 4 (2011), no. 4, 835–918. MR2860345

work page 2011

[13] [13]

, Gauge theory and Rasmussen ’s invariant , J. Topol. 6 (2013), no. 3, 659–674. MR3100886

work page 2013

[14] [14]

Lukas Lewark, Khovanov-Rozansky homologies, knotted weighted webs and t he slice genus , PhD thesis

work page

[15] [15]

, Rasmussen ’s spectral sequences and the slN -concordance invariants, Adv. Math. 260 (2014), 59–83. MR3209349

work page 2014

[16] [16]

Charles Livingston, Computations of the Ozsv´ ath-Szab´ o knot concordance invariant, Geom. Topol. 8 (2004), 735–742. MR2057779

work page 2004

[17] [17]

Andrew Lobb, Computable bounds for Rasmussen ’s concordance invariant , Compos. Math. 147 (2011), no. 2, 661–668. MR2776617

work page 2011

[18] [18]

Olga Plamenevskaya, Transverse knots and Khovanov homology , Math. Res. Lett. 13 (2006), no. 4, 571–586. MR2250492

work page 2006

[19] [19]

Jacob Rasmussen, Khovanov homology and the slice genus , Invent. Math. 182 (2010), no. 2, 419–447. MR2729272

work page 2010

[20] [20]

Lee Rudolph, Quasipositivity as an obstruction to sliceness , Bull. Amer. Math. Soc. (N.S.) 29 (1993), no. 1, 51–59. MR1193540

work page 1993

[21] [21]

Shumakovitch, Rasmussen invariant, slice-Bennequin inequality, and sli ceness of knots , J

Alexander N. Shumakovitch, Rasmussen invariant, slice-Bennequin inequality, and sli ceness of knots , J. Knot Theory Ramiﬁcations 16 (2007), no. 10, 1403–1412. MR2384833 Tokyo Institute of Technology, Ookayama, Meguro-ku, Tokyo Email address : iida.n.ad@m.titech.ac.jp

work page 2007