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The Jones-Krushkal polynomial and minimal diagrams of surface links

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Pith's one-line read The paper proves the first and second Tait conjectures for alternating links on surfaces and for virtual links, using a two-variable Jones-type polynomial.

desk verdict The KMT theorem for adequate surface diagrams is a real advance and the surface proof holds up, but the virtual-link Tait conjectures hang on an unproved minimal-genus assertion cited to preprints. read the letter →

arxiv 1908.06453 v1 pith:ACG5PGCX submitted 2019-08-18 math.GT

Hans U. Boden , Homayun Karimi This is my paper · ORCID
classification math.GT MSC 57M2557M27
keywords KauffmanbracketJones-KrushkalpolynomialadequatediagramalternatinglinkTaitconjecturesvirtualthickenedsurfacecrossingnumber
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 establishes the analogue of the classical Kauffman–Murasugi–Thistlethwaite theorem for links drawn on surfaces: any connected reduced alternating diagram of a link in a thickened surface has the fewest crossings among all diagrams of that link, and any two reduced alternating diagrams of the same oriented link have the same writhe. The same two conclusions are proved for all adequate diagrams, a wider class that includes every reduced alternating diagram, and are then transferred to virtual links. The engine is a two-variable Jones-type invariant whose span is controlled by the diagram's crossings and the surface's genus, so the crossing-counting argument becomes an inequality between polynomial degrees.

What carries the argument

The homological Kauffman bracket $\langle D\rangle_\Sigma$ is a state sum over the $2^n$ smoothings of $D$; a state $S$ contributes $A^{a(S)-b(S)}(-A^{-2}-A^2)^{k(S)}z^{r(S)}$, where $k(S)$ counts the cycles of $S$ that are null-homologous in $\Sigma$ and $r(S)$ is the rank of the map $H_1(S)\to H_1(\Sigma)$. The variable $z$ is what makes the invariant sensitive to the surface, and the Jones-Krushkal polynomial is its normalized, oriented version. Adequacy means that switching one smoothing away from the all-$A$ (or all-$B$) state never increases $k$; Proposition 2.8 shows every reduced alternating surface diagram is adequate, and the dual state lemma bounds $k(S)+k(S^\vee)$ for cellularly embedded diagrams. These ingredients combine into the span equalities that carry the proof.

What would settle it

A concrete check: take a reduced alternating diagram of a virtual link and look for a simple closed curve on the surface that is disjoint from the diagram and cuts off a handle; destabilizing along such a curve would produce a diagram of the same virtual link on a lower-genus surface, and if that diagram has fewer crossings, the theorem's minimality claim is false. Computationally, this is a finite search over the small virtual knot tables.

Watch

Extended reading notes

Core claim

The central claim is that a homological refinement of the Jones polynomial is strong enough to force crossing minimality in surfaces. For a diagram $D$ with $n$ crossings on a surface $\Sigma$ of genus $g$, the paper proves the span bound $\operatorname{span}(\langle D\rangle_\Sigma) \le 4n - 4g + 4$, with equality for adequate diagrams; for a connected reduced alternating diagram, equality holds and the span is exactly $4n - 4g + 4$. Because the span is invariant under Reidemeister moves, any diagram of the same link must have at least $n$ crossings, and a parallel-construction argument shows that two adequate diagrams of the same oriented link have the same writhe. For virtual links, the same results follow once reduced alternating diagrams are known to represent the virtual link on a minimal-genus surface.

Load-bearing premise

The load-bearing premise is that every reduced alternating diagram of a virtual link already sits on the lowest-genus surface representing that link, so that destabilization cannot yield a smaller-genus diagram of the same virtual link; if that premise fails, the span inequality that drives the virtual-link proof no longer follows.

Editorial extensions

If this is right

  • For a link in a thickened surface, any connected reduced alternating diagram gives its exact crossing number, so alternating surface links have a computable, diagram-independent crossing number.
  • Any two reduced alternating diagrams of the same oriented link in a thickened surface have equal writhe, so the writhe is a well-defined invariant for alternating surface links.
  • The same two conclusions hold for virtual links: reduced alternating virtual diagrams have minimal crossing number, and reduced alternating diagrams of the same virtual link share a writhe.
  • The minimality result is not limited to alternating diagrams: it applies to every adequate diagram on a fixed surface.
  • Because reduced alternating diagrams are adequate, the first and second Tait conjectures for surface links are consequences of a single span equality for the Jones-Krushkal polynomial.

Reading between the lines

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

  • A natural test of the virtual-link result is the paper's Conjecture 5.1: a computational search for a reduced alternating virtual diagram with a destabilizing curve would either confirm or refute the minimal-genus premise on which the transfer from surfaces to virtual links rests.
  • The adequacy condition is weaker than the classical plus/minus-adequacy, so the same span machinery should give crossing-number lower bounds for families of non-alternating surface diagrams whose all-$A$ and all-$B$ states can be controlled, such as positive surface diagrams.
  • Because the Jones-Krushkal polynomial is a homological refinement, a triply graded homology categorifying it would be a natural candidate for an invariant that detects virtual unknots, a task ordinary virtual Khovanov homology is known not to accomplish.
  • The span equality for reduced alternating surface diagrams has the same shape as the classical Jones-polynomial identity, which suggests asking whether equality on a surface also forces a sequence of flype moves; the paper explicitly leaves the flyping question open for virtual links.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper proves a Kauffman-Murasugi-Thistlethwaite theorem for alternating links in thickened surfaces: a connected reduced alternating diagram has minimal crossing number among diagrams on the same surface, and any two reduced alternating diagrams of the same oriented link have the same writhe. The result is stated more generally for adequate diagrams, using Krushkal's two-variable Jones-Krushkal polynomial and its associated homological Kauffman bracket. The main technical ingredients are a state-sum definition of the bracket, an adequacy criterion for diagrams on surfaces, the dual state lemma, and an r-parallel argument adapted from Stong. The paper then applies these results to virtual links, claiming the first and second Tait conjectures for reduced alternating virtual links (Theorem 5.3).

Significance. If the results are correct, this is a substantial extension of classical Tait theory to links in thickened surfaces and to virtual links, and it demonstrates that the Jones-Krushkal polynomial is a useful tool for crossing-number questions. The paper contains a clear state-sum setup, a number of nontrivial worked examples, a table of computations, and an explicit generalization of the dual state lemma. The surface-level main theorem is plausibly sound via the later Corollary 4.5 and the r-parallel argument, and the virtual-link extension is an important intended application. However, the virtual-link theorems currently depend on an unproved minimal-genus assertion imported from external and partially unpublished sources, and the direct proof of Theorem 4.1 has a gap involving the cellularity hypothesis of Lemma 2.10(b).

major comments (3)
  1. [Section 4, Theorem 4.1] The proof of Theorem 4.1 applies Lemma 2.10(b) to an arbitrary connected diagram D and concludes that span(⟨D⟩Σ) ≤ 4n + 4 − 4g by way of k(SA) + k(SB) ≤ n + 2 − 2g. Lemma 2.10(b) is stated only for cellularly embedded diagrams, and its proof uses that hypothesis to obtain surjectivity of j∗. Since Theorem 4.1 must apply to arbitrary competitor diagrams, which need not be cellularly embedded, the inequality is not justified as written. I note that Corollary 4.5, obtained through Theorem 4.4 and Lemma 4.3, gives a separate route to the same minimality and writhe statements on a fixed surface and does not use Lemma 2.10(b); the authors should either revise Theorem 4.1 to use that route or explicitly add and justify the cellularity hypothesis where needed.
  2. [Section 5, page 26, equations (15)-(18)] Theorem 5.3 depends on the unproved assertion that every reduced alternating diagram of a virtual link has minimal genus. The text states 'There are several ways to prove this' and cites [AARH+19], [AEG+19], and the unpublished preprint [BCK19] by two of the current authors, but no proof is supplied. This assertion is load-bearing: equation (17) applies Theorem 2.9 to D′′ on the minimal-genus surface Σ′, and equation (18) uses that equality to derive n′′ ≤ n. If some reduced alternating diagram D′′ could be destabilized to a lower-genus surface while preserving the virtual link type, then span(⟨D′′⟩Σ′) would not be 4(n′′−g′) + 4, and the conclusion n′′ ≤ n would not follow. The virtual-link theorem should either include a proof of the minimal-genus claim or be stated conditionally with the precise external result quoted.
  3. [Section 5, Theorem 5.2] Theorem 5.2 for virtual knots invokes Conjecture 5.1 as 'known to be true for virtual knots' and refers to Manturov [Man13], but the manuscript does not state the exact theorem from [Man13] that implies the needed claim. Because this conjecture is what lets the proof reduce a minimal crossing diagram to a minimal genus diagram, the statement should quote the relevant result precisely or prove it, so that Theorem 5.2 is independently verifiable from the cited source.
minor comments (5)
  1. [Introduction] There is a stray parenthesis in the sentence 'In [AFLT02]), Adams et al. use geometric methods...'; it should read 'In [AFLT02], Adams et al. ...'.
  2. [Example 3.5] The text says the virtual Hopf link diagram and its states are 'depicted in Figure 8', but Figure 8 shows the virtual trefoil; the intended reference appears to be Figure 13.
  3. [Corollary 4.5] The statements in Corollary 4.5 are relative to diagrams on one fixed surface Σ; making this explicit in the statements would prevent confusion with stable equivalence or virtual-link diagrams on surfaces of different genera.
  4. [Section 5, first paragraph] The claim that 'any reduced alternating diagram D of a virtual link L has minimal genus' is asserted for reduced diagrams, while the following sentence says 'any alternating virtual link diagram for L has minimal genus'; these differ if non-reduced alternating diagrams are allowed, so the intended hypothesis should be stated consistently.
  5. [References] The reference [BCK19] is an unpublished preprint by two of the current authors and is cited for a central statement in Section 5; if it remains essential, the relevant theorem from that preprint should be stated in the paper or the preprint should be made publicly available in a citable form.

Circularity Check

0 steps flagged · score 2.0 of 10

No constructional circularity; minor self-citations in the virtual-link extension are not load-bearing.

full rationale

The central derivation is a proof from Krushkal's externally defined homological Kauffman bracket, not a fit or normalization trick. The state sum (1), the definitions of A-/B-adequacy (Definition 2.5), and the span estimates (Lemma 2.6, Corollary 2.7) are structural; no parameter is fitted to the target crossing-number or writhe assertions. Proposition 2.8 and Theorem 2.9 show reduced alternating diagrams are adequate and compute their spans, and Corollary 4.5 deduces minimal crossing number and writhe equality from the span invariant. That chain is self-contained: the only noted difficulty, applying Lemma 2.10(b) to non-cellular diagrams in the proof of Theorem 4.1, is a verification gap, not a definitional reduction. The virtual-link extension in Section 5 imports the assertion that every reduced alternating virtual diagram has minimal genus ('There are several ways to prove this'), citing Adams et al. [AARH+19, AEG+19] and the authors' own unpublished preprint [BCK19]. That assertion is load-bearing for equations (16)-(18), and one cited route is a self-citation, but the same assertion is also supported by independent external results, so the argument does not reduce to a self-citation chain. The surface-link Tait theorem and its proof do not depend on [BCK19] or [Kar18]. Overall no prediction is equivalent to its input by construction; the score reflects only the minor self-citations in the virtual-link part.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The proof is a derivation from Krushkal's homological Kauffman bracket, plus imported geometric results. No free parameters or invented entities. The principal external dependencies are the minimal-genus claim for reduced alternating virtual link diagrams, supported partly by external preprints and partly by an unpublished self-cited preprint, and Kuperberg's uniqueness theorem for irreducible representatives.

assumptions (4)
  • domain assumption Krushkal's homological Kauffman bracket and its state-sum properties are invariant under regular isotopy of links in Sigma times I.
    The paper imports the bracket, including Lemmas 2.1 through 2.3, from Krushkal [Kru11] rather than proving them from first principles; all subsequent span arguments use these properties.
  • domain assumption Kuperberg's theorem: every stable equivalence class of links in thickened surfaces has a unique irreducible representative up to diffeomorphism.
    Used in Section 5 to identify minimal genus representatives so that spans of diagrams on different minimal genus surfaces can be compared.
  • domain assumption Any reduced alternating diagram of a virtual link has minimal genus.
    Attributed to [AARH+19], [AEG+19], and [BCK19]; the last is an unpublished preprint by two of the current authors. This is the principal external dependency for the virtual-link Tait conjectures.
  • standard math All homology groups use Z/2 coefficients, and cycles of a state span an isotropic subspace of H1(Sigma).
    Used throughout the definition of the homological bracket and in the dual state lemma.

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Pith. "Pith review of The Jones-Krushkal polynomial and minimal diagrams of surface links." pith.science (2026). https://pith.science/paper/ACG5PGCX

@misc{pith2026190806453,
  author       = {Pith},
  title        = {Pith review of: The Jones-Krushkal polynomial and minimal diagrams of surface links},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ACG5PGCX}},
  note         = {Machine review of arXiv:1908.06453}
}
read the original abstract

We prove a Kauffman-Murasugi-Thistlethwaite theorem for alternating links in thickened surfaces. It states that any reduced alternating diagram of a link in a thickened surface has minimal crossing number, and any two reduced alternating diagrams of the same link have the same writhe. This result is proved more generally for link diagrams that are adequate, and the proof involves a two-variable generalization of the Jones polynomial for surface links defined by Krushkal. The main result is used to establish the first and second Tait conjectures for links in thickened surfaces and for virtual links.

Figures

Figures reproduced from arXiv: 1908.06453 by the authors.

Figure 1
Figure 1. The detour move. Given a virtual link diagram D, the crossing number is denoted c(D) and is defined to be the number of classical crossings of D. The crossing number of a virtual link L is the minimum crossing number c(D) taken over all virtual link diagrams D representing L [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. The virtual trefoil, Hopf link, and Borromean rings. Given an oriented virtual link, each classical crossing is either positive or negative, see [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. A positive and a negative crossing. 1.2. Links in thickened surfaces. Virtual links can also be defined as equivalence classes of links in thickened surfaces. Let I = [0, 1] denote the unit interval and Σ be a compact, connected, oriented surface. A link in the thickened surface Σ × I is an embedding L: Fm i=1 S 1 ,→ Σ × I, considered up to isotopy and orientation preserving homeomorphisms of the pair (Σ × I, Σ × {0… view at source ↗
Figures from the paper (17 more)
Figure 4
Figure 4. Figure 4: Regular isotopy includes the above move together with Rei￾demeister 2 and 3 moves. and destabilizations. Stabilization is the operation of adding a handle to Σ to obtain a new surface Σ 0 , and destabilization is the opposite procedure. Specifically, if D0 and D1 are t…
Figure 5
Figure 5. Figure 5: A nugatory crossing. Definition 1.5. A surface link diagram D on Σ is called reduced if it is cellularly embedded and has no nugatory crossings. Remark 1.6. Note that, by the Kamada-Kamada construction, any virtual link can be realized by a cellularly embedded diagram …
Figure 6
Figure 6. Figure 6: An alternating knot diagram on the torus which is not checkerboard colorable. Several authors have used slightly different names for the notion of checkerboard col￾orability. For instance, in [KNS02], checkerboard colorable links are called normal, and in [Rus18], chec…
Figure 7
Figure 7. Figure 7: The A- and B-smoothing of a crossing. A state is a collection of simple closed curves on Σ which results from smoothing each of the crossings of D. Thus, a state S is just a link diagram on Σ with no crossings. Since there are two ways to smooth each crossing, there ar…
Figure 8
Figure 8. Figure 8: A minimal genus diagram of the virtual trefoil in the torus, and the states SA and SB. Example 2.4. The virtual trefoil K (see [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]
Figure 9
Figure 9. Figure 9: The three types of cobordisms from S to S 0 include a fusion (left), fission (middle), and single cycle smoothing (right). Further, in case (i), either r(S 0 ) = r(S) and k(S 0 ) = k(S) − 1 or r(S 0 ) = r(S) − 1 and k(S 0 ) = k(S); and in case (ii), either r(S 0 ) = r(…
Figure 10
Figure 10. Figure 10: The band β connecting the disk at ci to the disk at cj . Since the crossings are opposite (over/under), the band will be untwisted if the smoothings are the same (AA or BB) and twisted if the smoothings are opposite (AB or BA). Let β be a band connecting the disk at c…
Figure 11
Figure 11. Figure 11: A flat band and a half-twisted band [PITH_FULL_IMAGE:figures/full_fig_p015_11.png]
Figure 12
Figure 12. Figure 12: Oriented smoothings at positive and negative crossings. To complete the proof, we apply Equation 3 one more time to see that (−A) −3w(D) hD | SσiΣ = (−A) −3w(D) [PITH_FULL_IMAGE:figures/full_fig_p017_12.png]
Figure 13
Figure 13. Figure 13: A minimal genus diagram of the virtual Hopf link in the torus, and the states SA and SB. Example 3.5. The virtual Hopf link L (see [PITH_FULL_IMAGE:figures/full_fig_p018_13.png]
Figure 14
Figure 14. Figure 14: A minimal genus diagram of the virtual Borromean rings in the torus, and the states SA and SB. Therefore, hDiT = A3d 2 + 3Ad + 3A−1dz + A−3d, where d = −A2 − A−2 . Since D has writhe w = 3, it follows that JL(t, z) = t − 2t 2 − t 3 − 3t 5/2 z. 3.3. Calculations. In th…
Figure 15
Figure 15. Figure 15: From left to right, a virtual link with four components, a minimal genus representative in the torus, and the states SA and SB. Example 3.10. A virtual link L with four components along with a minimal genus diagram on the torus T appear on the left of [PITH_FULL_IMAG…
Figure 16
Figure 16. Figure 16: The virtual chain link. Example 3.11 [PITH_FULL_IMAGE:figures/full_fig_p021_16.png]
Figure 17
Figure 17. Figure 17: The states SA and SB for the virtual chain link. 3.4. Horizontal and vertical mirror images. In this section, we describe how the Jones-Krushkal polynomial changes under taking mirror images. Recall that there are two ways to take the mirror image of a virtual link L.…
Figure 18
Figure 18. Figure 18: The virtual knot K = 3.1 and its mirror images. Proposition 3.12. If L is a link in Σ × I, then JeL∗ (t, z) = JeL(t −1 , z), and Je L† (t, z) = JeL(t −1 , z). If L is a checkerboard colored link in Σ × I, then JL∗ (t, z) = JL(t −1 , z), and JL† (t, z) = JL(t −1 , z). …
Figure 19
Figure 19. Figure 19: A minimal genus diagram of 3.1 and the states SA and SB [PITH_FULL_IMAGE:figures/full_fig_p022_19.png]
Figure 20
Figure 20. Figure 20: The flype move for virtual links, where “T” is a classical tangle diagram. In a different direction, one can ask whether the Tait conjectures continue to hold in the welded category. Problem 5.6. Are the Tait conjectures true for alternating welded links? [PITH_FULL_…

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Works this paper leans on

7 extracted references · 7 canonical work pages

  1. [1]

    256 (2019), 262–278

    [AARH+19] Colin Adams, Carlos Albors-Riera, Beatrix Haddock, Zhiqi Li, Daishiro Nishida, Braeden Reinoso, and Luya Wang,Hyperbolicity of links in thickened surfaces, Topology Appl. 256 (2019), 262–278. MR3916014 [AEG+19] Colin Adams, Or Eisenberg, Jonah Greenberg, Kabir Kapoor, Zhen Liang, Kate O’Connor, Natalia Pacheco-Tallaj, and Yi Wang,Tg-hyperbolicit...

    work page 2019

  2. [3]

    THE JONES-KRUSHKAL POLYNOMIAL AND MINIMAL DIAGRAMS OF SURF ACE LINKS 31 [BGH+17] Hans U

    Preprint. THE JONES-KRUSHKAL POLYNOMIAL AND MINIMAL DIAGRAMS OF SURF ACE LINKS 31 [BGH+17] Hans U. Boden, Robin I. Gaudreau, Eric Harper, Andrew J. Nicas, and Lindsay White, Virtual knot groups and almost classical knots, Fundamenta Mathematicae138 (2017), 101–142. [CKS02] J. Scott Carter, Seiichi Kamada, and Masahico Saito, Stable equivalence of knots on...

    work page 2017

  3. [20]

    MR3105303 [MT93] William Menasco and Morwen Thistlethwaite, The classification of alternating links, Ann. of Math. (2)138 (1993), no. 1, 113–171. MR1230928 [Mur87] Kunio Murasugi, Jones polynomials and classical conjectures in knot theory, Topology 26 (1987), no. 2, 187–194. MR895570 32 H. U. BODEN AND H. KARIMI [Rus18] William Rushworth, Doubled Khovanov ...

    work page 1993

  4. [573]

    Knot Theory Ramifications17 (2008), no

    MR2343139 [CV08] Sergei Chmutov and Jeremy Voltz,Thistlethwaite’s theorem for virtual links, J. Knot Theory Ramifications17 (2008), no. 10, 1189–1198. MR2460170 [DK05] Heather A. Dye and Louis H. Kauffman,Minimal surface representations of virtual knots and links, Algebr. Geom. Topol.5 (2005), 509–535. MR2153118 [Dye17] Heather A. Dye,Checkerboard framings ...

    work page 2008

  5. [1997]

    Manturov,Kauffman-like polynomial and curves in 2-surfaces, J

    MR1472978 [Man03] Vassily O. Manturov,Kauffman-like polynomial and curves in 2-surfaces, J. Knot Theory Ramifications12 (2003), no. 8, 1145–1153. MR2017986 [Man04] , The Khovanov polynomial for virtual knots, Dokl. Akad. Nauk398 (2004), no. 1, 15–18. MR2128214 [Man07] , Khovanov’s homology for virtual knots with arbitrary coefficients, Izv. Ross. Akad. Nauk S...

    work page 2003

  6. [2018]

    Kauffman,State models and the Jones polynomial, Topology 26 (1987), no

    [Kau87] Louis H. Kauffman,State models and the Jones polynomial, Topology 26 (1987), no. 3, 395–407. MR899057 [Kau99] ,Virtual knot theory,EuropeanJ.Combin. 20(1999),no.7,663–690. MR1721925 (2000i:57011) [KK00] Naoko Kamada and Seiichi Kamada,Abstract link diagrams and virtual knots, J. Knot Theory Ramifications9 (2000), no. 1, 93–106. MR1749502 (2001h:5700...

    work page 1987

  7. [2019]

    TG-Hyperbolicity of Virtual Links

    ArXiv/1904.06385. [AFLT02] Colin Adams, Thomas Fleming, Michael Levin, and Ari M. Turner,Crossing number of alternating knots inS× I, Pacific J. Math.203 (2002), no. 1, 1–22. MR1895923 [Ban30] Carl Bankwitz, über die Torsionszahlen der alternierenden Knoten, Math. Ann. 103 (1930), no. 1, 145–161. MR1512619 [BCK19] Hans U. Boden, Micah Chrisman, and Homayun...

    work page Pith review arXiv 2002

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