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Thistlethwaite Theorems for Knotoids and Linkoids

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Pith's one-line read The arrow polynomial of a knotoid equals a single evaluation of the Bollobás-Riordan polynomial of a marked ribbon graph associated to any Kauffman state.

desk verdict Real Thistlethwaite extension for knotoids with a solid virtual case, but the twisted and planar theorems depend on an unproved partial-duality invariance. read the letter →

arxiv 2412.12357 v1 pith:M7FKGORL submitted 2024-12-16 math.GT

Sergei Chmutov , Qingying Deng , Joanna A. Ellis-Monaghan , Sergei Lando , Wout Moltmaker This is my paper · ORCID
classification math.GT MSC 57K1057K1205C31
keywords knotoidslinkoidstwistedarrowpolynomialloopBollobás-RiordanmarkedribbongraphsThistlethwaitetheorem
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 extends Thistlethwaite's theorem—the classical bridge from the Jones polynomial to graph theory—to knotoids, open-ended knot diagrams that arise in the study of protein topology. Its central claim is that the arrow polynomial of a knotoid, and its twisted and planar variants, can be recovered as a single evaluation of a Bollobás-Riordan polynomial of a marked ribbon graph built from any Kauffman state of the diagram. If true, this gives knotoid invariants a purely combinatorial core: instead of summing over diagram smoothings, one can sum over spanning subgraphs of a ribbon graph, and the choice of initial state is provably irrelevant. The paper also introduces twisted knotoids, which model knotoids on compact possibly non-orientable surfaces, and extends the identity to spherical linkoids with multiple components.

What carries the argument

The load-bearing construction is the marked ribbon graph $G_K^s$ associated to a Kauffman state $s$ of a knotoid diagram $K$: loops of the state become ordinary vertex disks, the unique arc component becomes a vertex disk with a marking that linearizes the cyclic order of its incident half-edges, and each smoothed crossing becomes a signed ribbon edge carrying two arrow decorations, with bars or a puncture carried along in the twisted and planar settings. The arrow Bollobás-Riordan polynomial $R_G$ sums over spanning subgraphs $F\subseteq E(G)$, recording the number of connected components, boundary components, and reduced arrow and bar decorations; the theorem evaluates it at $a=1$ with edge weights $b_e$ determined by the crossing signs. Partial duality is the mechanism that makes the theorem independent of the chosen state: passing from one state to another is exactly partial dualizing the marked ribbon graph, and $R_G$ transforms by a known product of edge weights under that operation.

What would settle it

A concrete calculation that would settle the central identity: take the one-crossing twisted knotoid of Example 3.15, compute its twisted arrow polynomial directly from the oriented state expansion, and compute the evaluation in Theorem 5.7 from the Bollobás-Riordan polynomial of its state ribbon graph; any mismatch in the coefficient of $\Lambda_1 K_1$ would refute the paper's central claim.

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Extended reading notes

Core claim

The central claim is Theorem 5.5: for a virtual knotoid diagram $K$ and any state $s$, the arrow bracket satisfies $\langle K\rangle = \frac{A^{e_+}B^{e_-}}{d}\, R_{G_K^s}(1,b,d)$, where $G_K^s$ is the marked ribbon graph whose vertices are the state components of $s$ and whose signed, arrow-decorated edges record the smoothed crossings. After substituting $B=A^{-1}$ and $d=-A^2-A^{-2}$, the normalized arrow polynomial becomes $A^{e_+-e_-}/(-A^2-A^{-2})$ times the arrow Bollobás-Riordan polynomial with edge weights $b_e=B/A$ or $A/B$. The same identity holds for twisted knotoids with bars (Theorem 5.7), for planar knotoids through the loop arrow polynomial (Theorem 5.8), and for spherical linkoids (Theorem 6.8), with the appropriate decorated version of the Bollobás-Riordan polynomial. Independence of the chosen state follows because changing states is exactly partial duality of marked ribbon graphs, and the relevant polynomial transforms by a known edge-weight factor under that operation.

Load-bearing premise

The geometric bridge identifying twisted knotoids with simple H-curves in thickened surfaces relies on the assertion, made without proof in Proposition 2.10, that any ambient isotopy between standard H-curves can be deformed so that the curve stays standard at every intermediate stage; if that smoothing step fails, the interpretation of twisted knotoids as knotoids on compact possibly non-orientable surfaces, and with it the stated significance of Theorems 5.7 and 5.8, would need revision.

Editorial extensions

If this is right

  • For any knotoid diagram, the arrow polynomial can be computed by enumerating spanning subgraphs of a single state's marked ribbon graph, rather than summing over all smoothings of the diagram.
  • The corollaries in Remark 5.6 and Corollary 5.9 recover Thistlethwaite-type statements for the Kauffman bracket of knotoids, twisted knotoids, and the Turaev loop bracket of planar knotoids when arrows and orientations are dropped.
  • Because Theorem 5.7 covers bars, the twisted arrow polynomial is invariant under stable equivalence of twisted knotoids, so the graph evaluation is a genuine invariant of knotoids on compact possibly non-orientable surfaces.
  • The linkoid version (Theorem 6.8) includes the odd-arrow, half-integer states that appear when several open components are present, so the same Thistlethwaite mechanism extends beyond single-arc diagrams.

Reading between the lines

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

  • Beyond the paper, the marked-ribbon-graph formulation suggests a practical route to tabulating knotoids: the Bollobás-Riordan evaluation can be computed from a single state, and state-independence via partial duality offers a built-in consistency check for tables such as the protein-motivated classifications.
  • One testable extension would be to use the paper's half-integer arrow states, introduced for linkoids, to define refined arrow polynomials for multi-component open diagrams and ask whether they distinguish linkoids that the current invariants do not separate.
  • The H-curve bijection, if it holds, also implies that the arrow polynomial of a planar knotoid can be reinterpreted as an invariant in a thickened sphere with a puncture, potentially connecting the loop arrow polynomial to existing invariants of tangles in such thickened surfaces.
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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

4 major / 4 minor

Summary. The paper introduces twisted knotoids as knotoid diagrams on compact, not necessarily orientable surfaces modulo stable equivalence, and associates to them a twisted arrow polynomial. It also defines marked ribbon graphs and three variants of the Bollobás–Riordan polynomial (arrow, twisted, and loop), and proves Thistlethwaite-type state-sum theorems: Theorem 5.5 for virtual/spherical knotoids, Theorem 5.7 for twisted knotoids, Theorem 5.8 for planar knotoids, and Theorem 6.8 for spherical linkoids. The main constructions are illustrated by explicit small examples.

Significance. If the gaps noted below are repaired, the paper would give a natural extension of Thistlethwaite's theorem to open-ended diagrams and would place the arrow polynomial of knotoids in a ribbon-graph framework. The definition of marked ribbon graphs and the state-sum correspondence in Theorem 5.5 are genuinely useful, and the worked examples are a strength. However, the advertised scope currently exceeds what is proved: several later theorems depend on partial-duality invariance statements that are only asserted by analogy, and the geometric interpretation of twisted knotoids rests on an unproved smoothing assertion.

major comments (4)
  1. [§5, Theorems 5.7 and 5.8; Proposition 4.13] The right-hand sides of Theorems 5.7 and 5.8 are claimed to be independent of the chosen state s, but the only partial-duality invariance proved in the paper is Proposition 4.13, and that is for the untwisted arrow polynomial R_G. The proofs of Theorems 5.7 and 5.8 are dismissed as 'Analogous' and 'again analogous'. This is a load-bearing omission: for Theorem 5.7, the polynomial R^t_G (Definition 4.7) involves bar decorations and the reduction rules of Figures 8 and 9, and Proposition 4.13 does not track bars through the partial-duality correspondence, where a bar can move from one boundary component to another when vertex and edge arcs are re-paired. For Theorem 5.8, R^ℓ_G (Definition 4.8) involves a puncture-nesting index ℓ whose behaviour under partial duality is not discussed at all. Until the partial-duality identities for R^t_G and R^ℓ_G are proved, or state-independence is proved directly, Theorems 5.7 and 5.8 are unsupported.
  2. [Proposition 2.10] The proof of Proposition 2.10 contains the statement 'It is also easy to see that any equivalence of standard H-curves can be modified such that the H-curve at every step of the deformation remains standard', with only a formal restatement following. This smoothing assertion is exactly what makes it legitimate to reduce H-curves to knotoid diagrams on Σ, and it is not obvious: an ambient isotopy between standard H-curves could temporarily move the rails away from being vertical fibres. Since Proposition 2.10 is the basis for the geometric interpretation of twisted knotoids and for the significance of Theorems 5.7 and 5.8, a complete proof or a precise reference is needed.
  3. [Lemmas 3.5 and 3.7] The classification of fully reduced twisted and planar states is only sketched. Lemma 3.5 asserts that after moving bars and cancelling arrows one always obtains one of the stated forms, but the interaction between Lemma 3.4 (passing a bar over an arrow flips the arrow) and the requirement of alternating arrows needs a detailed case analysis, especially for loop components with an odd number of bars. Lemma 3.7 asserts that all non-contractible loop components form a nested set, without ruling out non-nested configurations when several loops separate the arc component from the puncture. These lemmas define the variables that appear in the state sums and in the Bollobás–Riordan definitions, so they should be proved in full.
  4. [Definition 6.6 and Theorem 6.8] Theorem 6.8 is again stated with a proof 'analogous to that of Theorem 5.5'. In addition, Definition 6.6 uses an index ℓ(f) defined as the minimum of the numbers of arc components in the interior and exterior of a circular boundary component f. This is meaningful only after fixing an embedding of the boundary component in the sphere, whereas R^m_G is defined for abstract marked ribbon graphs. The manuscript should specify how ℓ(f) is computed from the ribbon graph data and how the partial-duality invariance needed for Theorem 6.8 is obtained.
minor comments (4)
  1. [Theorems 5.5, 5.7, 5.8] The displayed formulas such as '⟨K⟩ = Ae+Be− d R_{G^s_K}(1,b,d)' should read (A^{e_+}B^{e_-}/d) R_{G^s_K}(1,b,d); the division by d is missing in the displayed equation, although the subsequent text and examples use it.
  2. [Definition 4.8] The exponent ℓ in the formula for R^ℓ_G is never defined. The prose after the definition only says that the variables denote reduced state components; it should state precisely how the puncture and the number ℓ of enclosing loops are determined for a spanning subgraph F.
  3. [Example 5.10] In the computation for the twisted knotoid K2, the text says 'where e1, e2 are the edges of G^s_{K1} as indicated in Figure 20'; this should refer to G^s_{K2}.
  4. [Remark 3.16 and Remark 4.12] The product rules for Λ_i and Λ'_i under concatenation are stated, but the corresponding behaviour of the variables K_i (and of the half-index variables in the linkoid setting) is not discussed, so the claimed multiplicativity is incomplete.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the state-sum bijection in Theorem 5.5 is a genuine translation; the main risks are unproved 'analogous' cases, not circularity.

full rationale

The central Thistlethwaite-type relation (Theorem 5.5) is not circular. The arrow Bollobás-Riordan polynomial in Definition 4.6 is defined for marked ribbon graphs by a sum over spanning subgraphs, with formal boundary-component variables (K_i, Λ_i, Λ'_i) subject to the same arrow-reduction rules; those variables also occur in the knotoid arrow polynomial, which gives the theorem a by-construction flavor, but the graph polynomial has independent content (deletion-contraction, partial-duality behaviour) and the proof is a bijection between Kauffman states of K and subgraphs of G^s_K, with matching A/B powers checked rather than assumed. State-independence is delegated to Proposition 4.13, which cites the unmarked partial-duality result [3, Prop. 2.7]; that is a legitimate independent prior theorem (it does not assume knotoids), so the self-citation is not load-bearing in a circular way. The paper does contain omissions that should be weighed as correctness risks rather than circularity: Proposition 2.10 asserts, with 'It is also easy to see', that any equivalence of standard H-curves can be kept standard throughout; Theorems 5.7 and 5.8 are dismissed as 'Analogous', although they require partial-duality invariance for the twisted and loop Bollobás-Riordan polynomials that is not proved in the text; and Definition 4.8 leaves the index ℓ in R^ℓ_G imprecise. These are gaps in proof detail, not reductions of the theorems to their inputs. Accordingly, there is no significant circularity, and the score is low.

Assumptions & free parameters 0 free parameters · 3 assumptions · 3 invented entities

No free parameters fitted. The paper relies on standard background (Reidemeister moves, Kauffman state sums, Jordan curve theorem) and on its own new definitions. The main unproved input is the smoothing assertion in Proposition 2.10 and the importing of twisted link equivalence from [2].

assumptions (3)
  • standard math Jordan curve theorem: a simple closed curve in the plane separates the plane into an interior and an exterior region.
    Used in Lemma 3.6 and Lemma 3.7 to establish that reduced planar states have no loop arrows and that loops can only be nested around the arc component.
  • domain assumption The twisted Reidemeister moves (Figure 2) generate the same equivalence on twisted knotoid diagrams as knotoid diagrams on compact surfaces up to stable equivalence.
    Imported from the analogous result for twisted links in [2] and adapted in Section 2.2, including endpoint moves V0 and T0. This underlies the interpretation of twisted knotoids.
  • ad hoc to paper For any ambient isotopy between standard H-curves, there exists a smooth map F keeping the H-curves standard at every stage.
    Asserted without proof in Proposition 2.10 ('It is also easy to see'). It is needed for the bijection between twisted knotoids and simple H-curves.
invented entities (3)
  • Twisted knotoids
    purpose: Model knotoids on compact, possibly non-orientable surfaces, with stable equivalence; generalize virtual knotoids.
    Defined first in this paper (Definition 2.5). No falsifiable external handle; it is a new mathematical object.
  • Marked ribbon graphs
    purpose: Ribbon graphs with a marked vertex whose boundary is linearized, used to represent the arc component of a knotoid state and to define arrow Bollobás-Riordan polynomials.
    Defined in Section 4.1. A new combinatorial tool; no external evidence needed.
  • H-curves
    purpose: Embeddings of an H-shaped graph into the orientable thickening of a compact surface, used to encode twisted knotoids (Proposition 2.10).
    Defined in Definition 2.8. They are a geometric model, not an independently observed entity.

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Pith. "Pith review of Thistlethwaite Theorems for Knotoids and Linkoids." pith.science (2026). https://pith.science/paper/M7FKGORL

@misc{pith2026241212357,
  author       = {Pith},
  title        = {Pith review of: Thistlethwaite Theorems for Knotoids and Linkoids},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M7FKGORL}},
  note         = {Machine review of arXiv:2412.12357}
}
read the original abstract

The classical Thistlethwaite theorem for links can be phrased as asserting that the Kauffman bracket of a link can be obtained from an evaluation of the Bollob\'as-Riordan polynomial of a ribbon graph associated to one of the link's Kauffman states. In this paper, we extend this result to knotoids, which are a generalization of knots that naturally arises in the study of protein topology. Specifically we extend the Thistlethwaite theorem to the twisted arrow polynomial of knotoids, which is an invariant of knotoids on compact, not necessarily orientable, surfaces. To this end, we define twisted knotoids, marked ribbon graphs, and their arrow- and Bollob\'as-Riordan polynomials. We also extend the Thistlethwaite theorem to the loop arrow polynomial of knotoids in the plane, and to spherical linkoids.

Figures

Figures reproduced from arXiv: 2412.12357 by the authors.

Figure 1
Figure 1. The one-crossing knotoid is trivial on S 2 . Remark 2.4. Example 2.3 exhibits precisely the difference between planar and spherical knotoids: the ability to move arcs past ∞. Thus spherical knotoids are equivalent to knotoids in the plane modulo spherical moves which move an exterior arc to the other side of the diagram [21]. Equivalently we can see planar knotoids as spherical knotoids containing a point marked ‘∞’… view at source ↗
Figure 2
Figure 2. The twisted Reidemeister moves. To see that twisted knotoid diagrams modulo the equivalence generated by the moves in [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Rules for thickening a twisted knotoid diagram (left) and an example of the [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (21 more)
Figure 4
Figure 4. Figure 4: A stabilization move on a thickened surface. [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: An H-curve that is neither simple nor standard (left), a simple non-standard H-curve (middle), and an equivalent standard H-curve (right). Let π be the map sending a standard H-curve in Σ×eI to a knotoid in Σ by projecting Σ × I onto the 0-section of Σ×eI. This indeed …
Figure 6
Figure 6. Figure 6: Image of [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 7
Figure 7. Figure 7: The oriented state expansion. The long black arrows indicate the original ‘tail [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]
Figure 8
Figure 8. Figure 8: Reduction rule for arrows. Lemma 3.2. The number of arrows on each state component of a state is even. Proof. Any state consists of a tuple of loop components and a single arc component. Picking a point x on an arbitrary loop component, consider the orientation of the …
Figure 9
Figure 9. Figure 9: Reduction rules for bars. Lemma 3.4. The reduction rules imply that we may pass a bar over an arrow, changing the arrow’s direction in the process. See [PITH_FULL_IMAGE:figures/full_fig_p009_9.png]
Figure 10
Figure 10. Figure 10: Moving a bar over an arrow. Lemma 3.5. Let C be a reduced state component in a twisted knotoid state. Then if C has arrows it is equivalent to one of {Ki ,Λi}i∈N, or else C is equivalent to a loop with a single bar on it, which we label by a variable K1/2. Here Ki ,Λi…
Figure 11
Figure 11. Figure 11: Example spherical knotoid K and its states. Next we extend the arrow polynomial to twisted knotoids: Definition 3.13. Let K be a twisted knotoid diagram. The twisted arrow bracket ⟨K⟩ t of K is the unique polynomial satisfying the same rules as in Definition 3.8, exce…
Figure 12
Figure 12. Figure 12: Example twisted knotoid K and its states. Remark 3.16. There is a natural operation of concatenation for knotoids and virtual knotoids. It is defined for diagrams and consists in attaching the head of the first knotoid to the tail of the second one, so that the tail o…
Figure 13
Figure 13. Figure 13: Example planar knotoid K and its states. 4.1 Marked ribbon graphs Definition 4.1. A ribbon graph G = (V (G), E(G)) is a surface with boundary consisting of a union of two sets of disks, a set V (G) of vertices and a set E(G) of edges, such that: 1. The vertices and ed…
Figure 14
Figure 14. Figure 14: A ribbon graph diagram. sign of every edge incident to that vertex. The bijection from ribbon graphs to rotation systems is then given by sending a ribbon graph diagram to its underlying graph, with the cyclic ordering being that of the edge ribbons on the original ve…
Figure 15
Figure 15. Figure 15: A marked ribbon graph diagram (left) and another decorated with signs, [PITH_FULL_IMAGE:figures/full_fig_p016_15.png]
Figure 16
Figure 16. Figure 16: If D ⊆ E(G) is a subset of edges of G then the partial dual GD of G with respect to D is formed by carrying out the same local replacement for each e ∈ D. As terminology suggests partial duality is a self-dual operation, in the sense that (GD) D = G for all D ⊆ E(G), …
Figure 16
Figure 16. Figure 16: Partial duality with respect to an edge marked with arrows. The vertices may [PITH_FULL_IMAGE:figures/full_fig_p017_16.png]
Figure 17
Figure 17. Figure 17: An example marked ribbon graph G with arrow decorations, its subgraphs, and their contributions. Next we formulate the contraction-deletion relation for the arrow Bollob´as-Riordan polynomial for marked ribbon graphs decorated with arrows. Here deletion and contrac￾ti…
Figure 18
Figure 18. Figure 18: A marked ribbon graph Gs K associated to a state s of a twisted knotoid K. Proposition 5.2. Let K be a twisted or planar knotoid and let s be a state of K. Then a diagram for K can be recovered from Gs K. Proof. Starting with a diagram for Gs K, contract its edges alo…
Figure 19
Figure 19. Figure 19: Associating a ribbon graph to a state of [PITH_FULL_IMAGE:figures/full_fig_p024_19.png]
Figure 20
Figure 20. Figure 20: Associating a ribbon graph to a state of [PITH_FULL_IMAGE:figures/full_fig_p024_20.png]
Figure 21
Figure 21. Figure 21: Associating a ribbon graph to a state of [PITH_FULL_IMAGE:figures/full_fig_p025_21.png]
Figure 22
Figure 22. Figure 22: A linkoid (left) and a multiknotoid (right). [PITH_FULL_IMAGE:figures/full_fig_p025_22.png]
Figure 23
Figure 23. Figure 23: A counterexample to Lemmas 3.2 and 3.6 for linkoids. [PITH_FULL_IMAGE:figures/full_fig_p026_23.png]

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