Equivariant Unknotting Number and Involutive Khovanov Homology

KeeTaek Kim

arxiv: 2604.08981 · v1 · submitted 2026-04-10 · 🧮 math.GT

Equivariant Unknotting Number and Involutive Khovanov Homology

KeeTaek Kim This is my paper

Pith reviewed 2026-05-10 17:03 UTC · model grok-4.3

classification 🧮 math.GT

keywords equivariant unknotting numberinvolutive Bar-Natan homologystrongly invertible knotsH-torsion orderKhovanov homologyunknotting numberknot invariants

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The pith

The equivariant unknotting number of a strongly invertible knot is bounded below by the H-torsion order of its involutive Bar-Natan homology.

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

This paper proves that for any strongly invertible knot the equivariant unknotting number is at least the H-torsion order taken from the involutive version of Bar-Natan homology. The result extends an earlier non-equivariant lower bound by Alishahi to the setting where the knot carries a strong inversion. A reader would care because the new bound is computable from homology and detects when preserving the symmetry forces extra unknotting moves. The authors apply it to all prime knots up to nine crossings and locate five examples where the ordinary unknotting number is strictly smaller than the equivariant one.

Core claim

We demonstrate that the equivariant unknotting number tilde u(K) of a strongly invertible knot K is bounded below by the H-torsion order tilde ord(K) of the involutive Bar-Natan homology tilde BN(K). This serves as an equivariant analogue to the bound established by Alishahi. As an application we identify five strongly invertible prime knots with crossing numbers at most 9 for which the strict inequality u(K) < tilde u(K) holds.

What carries the argument

The involutive Bar-Natan homology tilde BN(K) of a strongly invertible knot, from which the H-torsion order tilde ord(K) is read off to serve as a lower bound on the equivariant unknotting number.

If this is right

The inequality tilde u(K) greater than or equal to tilde ord(K) holds for every strongly invertible knot.
The bound recovers Alishahi's result in the non-equivariant limit.
At least five prime knots of crossing number nine or less satisfy u(K) strictly less than tilde u(K).
Homology computations now yield concrete lower bounds on the number of symmetric unknotting moves.

Where Pith is reading between the lines

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

The new bound may allow systematic enumeration of minimal equivariant unknotting sequences for all small strongly invertible knots.
Symmetry can increase the minimal number of crossing changes needed to reach the unknot for some knots.
Pairing this torsion-order bound with other equivariant invariants could produce exact values of tilde u(K) for many knots.

Load-bearing premise

The involutive Bar-Natan homology is well-defined for strongly invertible knots and its H-torsion order supplies a correct lower bound on the equivariant unknotting number.

What would settle it

A strongly invertible knot K for which the actual equivariant unknotting number is smaller than the H-torsion order computed from its involutive Bar-Natan homology.

Figures

Figures reproduced from arXiv: 2604.08981 by KeeTaek Kim.

**Figure 1.** Figure 1: Examples of morphisms Σ1, Σ2 : O1 ⊔ O2 ⊔ O3 → O′ 1 ⊔ O′ 2 in the category Cob•. without loss of generality that the involution τ of an involutive link (K, τ ) is given by a 180-degree rotation about the y-axis (viewing S 3 as the one-point compactification of R 3 ). In this setting, a knot diagram projected onto the xy-plane is called a transvergent diagram. We refer to the set of fixed points of the rotat… view at source ↗

**Figure 2.** Figure 2: Resolution of a crossing. In the following paragraphs, we will define a chain complex JDKK in Kob• for a given diagram DK of a link K, and it turns out to be a knot invariant, which will be denoted by JKK in the category Kob•/h. Fix a diagram DK of a link K, and suppose DK has n crossings. We enumerate the crossings of DK from 1 to n. We can resolve each crossing in two ways, called 0-resolution and 1-reso… view at source ↗

**Figure 3.** Figure 3: A formal Khovanov complex for the left-hand trefoi [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 4.** Figure 4: Building blocks for 2-dimensional (dotted) cobor [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

**Figure 5.** Figure 5: Additional relations for Cob•. Consequently, we obtain a functor from Cob• to (F[H] − Mod). One can verify that this functor satisfies the four relations provided in Theorem 3. It therefore descends to a functor on Cob•/l, and subsequently to the categories of complexes Kob• and Kob•/h. By applying this TQFT to JKK, we obtain a chain complex CBN(K) of F[H]-modules whose chain homotopy type is an invariant … view at source ↗

**Figure 6.** Figure 6: Three types of equivariant crossing change. [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗

**Figure 7.** Figure 7: Local pictures of diagrams K, K′ , K0, and K1. Proof of Theorem 1. The theorem follows from Theorem 11, Theorem 13, and the fact that gord(U) = 0. Remark. In general, the same proof establishes a lower bound for the equivariant Gordian distance ˜d(K, J) between two given strongly invertible knots K and J, defined as the minimal number of self-intersections occurring during an involution-invariant homotopy… view at source ↗

**Figure 8.** Figure 8: The chain map D. Define maps JKK f0 −↽⇀− g0 JK′ K using the following diagram: JKK = JK0K JK1K JK′ K = JK1K JK0K f0 S id g0 D S id D Here, D is a cobordism map consisting of the sum of two cylinders, each with a dot placed on either side of the crossing c (see [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗

**Figure 9.** Figure 9: A diagrammatic description of the construction of [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗

**Figure 10.** Figure 10: Local pictures of diagrams K, K′ , and their resolutions. Each vertical line denotes the axis. Note that the involution Iτ maps JKij K → JKjiK for i, j ∈ {0, 1}. We define two chain maps JKK f −↽⇀− g JK′ K by f =     0 0 0 DlDr 0 0 Dr 0 0 Dl 0 0 id 0 0 0     , g =     0 0 0 id 0 0 Dl 0 0 Dr 0 0 DlDr 0 0 0     , where we decompose JKK as JK00K ⊕ JK10K ⊕ JK01K ⊕ JK11K and JK′ K as JK11K ⊕ JK… view at source ↗

**Figure 11.** Figure 11: (Type A) Two chain maps JKK f −↽⇀− g JK′ K. 16 [PITH_FULL_IMAGE:figures/full_fig_p016_11.png] view at source ↗

**Figure 12.** Figure 12: (Type A) Red arrow depicts the map gf + H2 · id, and green dashed arrow depicts the homotopy h. JK10K JKK = JK00K JK11K JK01K JK10K JKK = JK00K JK11K JK01K Sl Sr Sr Sl SlSrIτ Iτ (H·Sl)+SrDlIτ Sl Sr Sr Sl IτSrDl+(H·Sl)Iτ Iτ (H·Sl)+SrDlIτ IτSrDl+(H·Sl)Iτ Iτh+hIτ [PITH_FULL_IMAGE:figures/full_fig_p017_12.png] view at source ↗

**Figure 13.** Figure 13: (Type A) Red arrow depicts the map Iτh+ hIτ , and green dashed arrow depicts the homotopy k. 17 [PITH_FULL_IMAGE:figures/full_fig_p017_13.png] view at source ↗

**Figure 14.** Figure 14: Local pictures of diagrams K, K′ and its resolution. Each vertical line denotes the axis. 3.3 Type C equivariant crossing change Let K′ be a transvergent diagram with two nearby fixed points, as shown in [PITH_FULL_IMAGE:figures/full_fig_p019_14.png] view at source ↗

**Figure 15.** Figure 15: Local pictures of diagrams K, K′ and its resolution K00, K10, K01, and K11. Each vertical line denotes the axis. K′ 11 is a diagram obtained from K11 by removing the circle. JK10K JKK = JK00K JK11K JK01K JK′ K = JK00K Sr f Sl Sr id α=SǫDl Sl g D ιS [PITH_FULL_IMAGE:figures/full_fig_p020_15.png] view at source ↗

**Figure 16.** Figure 16: (Type C) Two chain maps JKK f −↽⇀− g JK′ K. + (a) The morphism α = SǫDl . + = (b) The morphism αιS = S 2 [PITH_FULL_IMAGE:figures/full_fig_p020_16.png] view at source ↗

**Figure 17.** Figure 17: Morphisms α and αιS. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_17.png] view at source ↗

**Figure 18.** Figure 18: (Type C) gf is drawn by red arrows. Now consider gf : JKK → JKK: gf =     D 0 0 Dα 0 0 0 0 0 0 0 0 ιS 0 0 ιSα     , as in [PITH_FULL_IMAGE:figures/full_fig_p021_18.png] view at source ↗

**Figure 19.** Figure 19: (Type C) Red arrow depicts the map gf +H ·id, and green dashed arrow depicts the homotopy h. JK11K JKK = JK10K JK01K JK00K JK11K JKK = JK10K JK01K JK00K Sl Sr Sr Sl IτSl+J −1 r ǫDlIτ Iτ J −1 r ǫDl+SlIτ ιJlIτ Iτ ιJl ιǫIτ SǫIτ IτSl SlIτ Sl Sr Sr Sl Iτh+hIτ [PITH_FULL_IMAGE:figures/full_fig_p022_19.png] view at source ↗

**Figure 20.** Figure 20: (Type C) Red arrow depicts the map Iτh+ hIτ , and green dashed arrow depicts the homotopy h. Caution: the complex JKK is represented in a different angle than previous diagrams. 22 [PITH_FULL_IMAGE:figures/full_fig_p022_20.png] view at source ↗

**Figure 21.** Figure 21: ιS2 ǫ + SrJ −1 r ǫ + id = ιJlSl . While the homotopy h is formally obtained by composing the homotopies from a Reidemeister move II with Alishahi’s map h0, we provide the direct computation here for clarity. We have to show gf + H · id =     D + H · id 0 0 0 0 H · id 0 0 0 0 H · id 0 ιS 0 0 ιSα + H · id     =     S 2 l 0 0 0 0 J −1 r ǫDlSr + S 2 l J −1 r ǫDlSl 0 0 0 S 2 l 0 ιJlSr 0 0 SrJ −1 r… view at source ↗

**Figure 22.** Figure 22: Sr + J −1 r ǫDl = SlSǫ. and it is represented as green dashed arrows in [PITH_FULL_IMAGE:figures/full_fig_p024_22.png] view at source ↗

**Figure 23.** Figure 23: Examples of prime knots K with u(K) = 1 < 2 ≤ ue(K). These knots are 77b, 821a, 928a, 934, and 939, in sequence. Appendix: table of invariants We have tabulated the following invariants for prime, strongly invertible knots with crossing numbers up to 9: (1) the maximal H-torsion order ord(K), (2) the Rasmussen invariant s(K), (3) the equivariant Rasmussen invariants s(K) and s(K), and (4) the equivariant … view at source ↗

read the original abstract

We demonstrate that the equivariant unknotting number $\widetilde{u}(K)$ of a strongly invertible knot $K$ is bounded below by the $H$-torsion order $\widetilde{\mathrm{ord}}(K)$ of the involutive Bar-Natan homology $\widetilde{\mathrm{BN}}(K)$. This result serves as an equivariant analogue to the bound established by Alishahi. As an application, we identify five strongly invertible prime knots with crossing numbers at most $9$ for which the strict inequality $u(K) < \widetilde{u}(K)$ holds.

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

The paper gives an equivariant version of Alishahi's unknotting bound and checks it on five small knots showing the numbers can differ.

read the letter

The paper shows that for strongly invertible knots, the equivariant unknotting number is at least the H-torsion order in the involutive Bar-Natan homology. It also points out five prime knots with nine or fewer crossings where the regular unknotting number is strictly less than this equivariant version. The new part is the equivariant lift of the bound and the concrete examples that demonstrate the two unknotting numbers are not always the same. The author takes the involutive homology setup and shows it gives a lower bound that respects the symmetry of the knot. This works because the homology is already set up to handle the involution, and the torsion order obstructs unknotting moves that preserve the inversion. The examples are the main payoff, as they turn the abstract bound into something you can see on actual diagrams. The soft spots are minor. The proof likely follows the non-equivariant case closely, so the main work is confirming that the homology remains invariant under the equivariant Reidemeister moves and that the torsion calculation is correct for those five knots. If the paper includes the diagrams or the computation steps, that would strengthen it. The result stays within the world of knot homology and geometric topology for symmetric knots. It does not claim to solve a big open problem or apply to all knots. A reader who follows papers on Khovanov homology and its variants would get value from knowing this bound and the examples. It is not essential for everyone in knot theory. The paper has a clear statement and some explicit content, so it deserves to go to a serious referee who can verify the calculations and the adaptation of the homology. I recommend sending it for peer review.

Referee Report

1 major / 3 minor

Summary. The manuscript proves that for a strongly invertible knot K the equivariant unknotting number tilde u(K) is bounded below by the H-torsion order tilde ord(K) of the involutive Bar-Natan homology tilde BN(K). This is presented as an equivariant lift of Alishahi's non-equivariant bound u(K) >= ord(K). The paper applies the inequality to five strongly invertible prime knots of crossing number at most 9, exhibiting cases where the ordinary unknotting number is strictly smaller than the equivariant unknotting number.

Significance. If the central inequality holds, the result supplies a computable homological obstruction to equivariant unknotting that is unavailable from ordinary Khovanov or Bar-Natan homology. The explicit examples demonstrate that the bound is sharp enough to detect knots for which symmetry forces additional unknotting moves, thereby furnishing concrete evidence that equivariant and non-equivariant unknotting numbers can differ. The construction re-uses an established homology theory rather than introducing new parameters, which strengthens its utility for further computations in the equivariant setting.

major comments (1)

[§3 (Main Theorem)] The manuscript asserts that tilde BN(K) is invariant under equivariant Reidemeister moves and that its H-torsion order decreases under equivariant unknotting sequences, but the precise chain-level argument extending Alishahi's proof is not expanded with explicit diagram moves or a verification that the involution commutes with the relevant differentials. This step is load-bearing for the claimed inequality.

minor comments (3)

[§4 (Applications)] The five knots in the application should be identified by their standard names or Rolfsen numbers together with a brief indication of the diagrams used to compute tilde ord(K).
[§2 (Preliminaries)] Notation for the involutive complex and the torsion order is introduced only after the statement of the main theorem; moving the definitions to §2 would improve readability.
[§4 (Applications)] A short table summarizing the computed values of u(K), tilde u(K) and tilde ord(K) for the five examples would make the strict inequality immediately visible.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive assessment of its significance. We address the single major comment below and have revised the manuscript accordingly to strengthen the exposition of the proof.

read point-by-point responses

Referee: [§3 (Main Theorem)] The manuscript asserts that tilde BN(K) is invariant under equivariant Reidemeister moves and that its H-torsion order decreases under equivariant unknotting sequences, but the precise chain-level argument extending Alishahi's proof is not expanded with explicit diagram moves or a verification that the involution commutes with the relevant differentials. This step is load-bearing for the claimed inequality.

Authors: We agree that the chain-level details in Section 3 were not sufficiently explicit. In the revised manuscript we have expanded this section with a new subsection that provides explicit verification for each equivariant Reidemeister move. For each move we exhibit the chain maps on the Bar-Natan complex, confirm that they intertwine with the involution, and show that the induced maps on homology preserve the H-torsion order. The argument for the decrease under equivariant unknotting sequences is likewise written out at the chain level, following the same filtration and cancellation steps as in Alishahi's work but with the involution action tracked throughout. These additions make the extension of the non-equivariant proof fully rigorous and self-contained. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is a direct equivariant extension of an external result

full rationale

The paper asserts a lower bound tilde u(K) >= tilde ord(K) by constructing an involutive Bar-Natan complex for strongly invertible knots and lifting Alishahi's non-equivariant argument. The abstract and claim structure present this as a proof resting on the well-definedness of the involutive homology and the obstruction property of its H-torsion order. No self-definitional loop, fitted parameter renamed as prediction, or load-bearing self-citation chain appears; the cited Alishahi result is external and the equivariant adaptation is asserted to follow by direct modification of the complex. The manuscript is therefore self-contained against external benchmarks with no reduction of the central inequality to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract only; the result relies on the pre-existing definition and properties of involutive Bar-Natan homology and on the notion of strong invertibility, none of which are introduced or fitted in the paper itself.

axioms (1)

domain assumption Involutive Bar-Natan homology is defined and functorial for strongly invertible knots
The bound presupposes the standard construction of tilde BN(K) and its H-torsion order.

pith-pipeline@v0.9.0 · 5385 in / 1272 out tokens · 54166 ms · 2026-05-10T17:03:06.639836+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages

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Unknotting number and Khovanov ho mology

[Ali19] Akram Alishahi. “Unknotting number and Khovanov ho mology”. In: Paciﬁc J. Math. 301.1 (2019), pp. 15–29. [Bar05] Dror Bar-Natan. “Khovanov’s homology for tangles a nd cobordisms”. In: Geom. Topol. 9 (2005), pp. 1443–1499. [Bor+25] Maciej Borodzik, Irving Dai, Abhishek Mallick, an d Matthew Stoﬀregen. Kho- vanov homology and equivariant surfaces

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Khovanov homology and equivariant surfaces.arXiv preprint arXiv:2507.13642, 2025

arXiv: 2507.13642 [math.GT] . [BC26] Keegan Boyle and Wenzhao Chen. “Equivariant unknott ing numbers of strongly invertible knots”. In: J. Lond. Math. Soc. (2) 113.3 (2026), Paper No. e70506. [BI22] Keegan Boyle and Ahmad Issa. “Equivariant 4-genera o f strongly invertible and periodic knots”. In: J. Topol. 15.3 (2022), pp. 1635–1674. [CY26] Daren Chen an...

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Equivariant knots and knot Floer homology

arXiv: 2506.00824 [math.GT] . [DMS23] Irving Dai, Abhishek Mallick, and Matthew Stoﬀregen . “Equivariant knots and knot Floer homology”. In: J. Topol. 16.3 (2023), pp. 1167–1236. [ILM25] Damian Iltgen, Lukas Lewark, and Laura Marino. “Kho vanov homology and rational unknotting”. In: Quantum Topol. 16.4 (2025), pp. 655–741. [Kho00] Mikhail Khovanov. “A cat...

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[Koc04] Joachim Kock

arXiv: 2503.05414 [math.GT] . [Koc04] Joachim Kock. Frobenius algebras and 2D topological quantum ﬁeld theorie s. Vol

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Cambri dge University Press, Cambridge, 2004, pp

London Mathematical Society Student Texts. Cambri dge University Press, Cambridge, 2004, pp. xiv+240. [KWZ19] Artem Kotelskiy, Liam Watson, and Claudius Zibrowi us. Immersed curves in Khovanov homology

work page 2004

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[LMZ24] Lukas Lewark, Laura Marino, and Claudius Zibrowius

arXiv: 1910.14584 [math.GT] . [LMZ24] Lukas Lewark, Laura Marino, and Claudius Zibrowius . Khovanov homology and reﬁned bounds for Gordian distances

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[LM26] Charles Livingston and Allison H

arXiv: 2409.05743 [math.GT] . [LM26] Charles Livingston and Allison H. Moore. KnotInfo: Table of Knot Invariants . URL: knotinfo.org. Jan

work page arXiv

[8] [8]

A reﬁnement of Khovanov homology

[L W21] Andrew Lobb and Liam Watson. “A reﬁnement of Khovanov homology”. In: Geom. Topol. 25.4 (2021), pp. 1861–1917. [MP23] Allison N. Miller and Mark Powell. “Strongly inverti ble knots, equivariant slice genera, and an equivariant algebraic concordance group”. I n: J. Lond. Math. Soc. (2) 107.6 (2023), pp. 2025–2053. 29 [MB84] John W. Morgan and Hyman ...

work page 2021

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The Conway knot is not slice

Academic Press, Inc., Orlando, FL, 1984, pp. xv+243. [Pic20] Lisa Piccirillo. “The Conway knot is not slice”. In: Ann. of Math. (2) 191.2 (2020), pp. 581–591. [Ras10] Jacob Rasmussen. “Khovanov homology and the slice g enus”. In: Invent. Math. 182.2 (2010), pp. 419–447. [Sak86] Makoto Sakuma. “On strongly invertible knots”. In: Algebraic and topological t...

work page 1984

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Involutive Khovanov homology and equ ivariant knots

Kinokuniya, Tokyo, 1986, pp. 176–196. [San25] Taketo Sano. “Involutive Khovanov homology and equ ivariant knots”. In: Algebr. Geom. Topol. 25.8 (2025), pp. 5059–5111. [YUI] Taketo Sano. YUI, https://github.com/taketo1024/yui. Version 0.4.0. [Wal69] Friedhelm Waldhausen. “ ¨Uber Involutionen der 3-Sph¨ are”. In:Topology 8 (1969), pp. 81–91. 30

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