{"id":"da5700e7-a774-4ba9-87ec-78ffc25e7a22","arxiv_id":"2412.12357","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The arrow, twisted arrow, and loop arrow polynomials of knotoids are shown to be evaluations of Bollobás-Riordan polynomials of associated marked ribbon graphs.","lead":"This paper proves Thistlethwaite theorems for knotoids: the arrow polynomial of a knotoid, and its twisted and loop variants, equals an evaluation of a Bollobás-Riordan polynomial of a marked ribbon graph built from a Kauffman state. This connects knotoid invariants, which are used to study protein topology, to graph polynomials.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Partial-duality invariance for the twisted and loop Bollobás-Riordan polynomials is asserted without proof; Theorems 5.7 and 5.8 depend on it for state-independence.","rationale":"I looked for the weakest step in the chain leading to the main theorems. The reader's candidate, the smoothing claim in Proposition 2.10, is indeed a gap: the proof asserts that any isotopy between standard H-curves can be made standard throughout, with only the sentence 'It is also easy to see' as support. But that claim underpins the geometric interpretation of twisted knotoids as knotoids on compact surfaces, not the algebraic content of the Thistlethwaite formulas. Even if Proposition 2.10 needed revision, Theorems 5.5, 5.7, 5.8, and 6.8 are statements about diagrams and decorated ribbon graphs and could still be correct. The more load-bearing gap is the transfer of the state-independence argument from Theorem 5.5 to its twisted and planar analogues. Theorem 5.5 is proved by combining Proposition 5.4 with the partial-duality invariance of R_G in Proposition 4.13. The latter is specific to arrow decorations; no analogue is stated, let alone proved, for R^t_G or R^ℓ_G. The obstruction is concrete: bars live on vertex boundaries, and partial duality changes the gluing of boundary arcs, so a bar's position relative to a given boundary component is not fixed; the loop polynomial has the additional unaddressed issue that the index ℓ is not even defined in Definition 4.8. These are not aesthetic complaints about omitted details: without partial-duality invariance, the right-hand side of Theorems 5.7 and 5.8 depends on the arbitrary state s used to build G_s^K, while the left-hand side does not. The proposed computation on Example 3.15's second state is a fast way to see whether the concern is real or merely a missing proof. I therefore keep the reader's conditional verdict, but for a different reason.","tokens_in":19825,"tokens_out":16243,"duration_ms":154844,"concrete_test":"Take the twisted knotoid K2 of Example 3.15 and compute the all-negative state s′ (equivalently, form G_{s′} = (G_s)^{{e1,e2}} from the all-positive G_s in Figure 20). Evaluate R^t_{G_{s′}}(1,b,d) directly from Definition 4.7 using the reduction rules of Figures 8 and 9, and compare with the value required by Theorem 5.7, namely (A^{e_+-e_-}/d)⟨K2⟩^t_A. If the two differ, Theorem 5.7 is false; if they agree, repeat the same comparison for a marked ribbon graph with one edge and one bar to test the general partial-duality identity R^t_G(1,b,c) = b_e R^t_{G^e}(1,1/b_e,c).","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Theorem 5.5 derives state-independence from Proposition 5.4 plus Proposition 4.13. Proposition 4.13 is proved only for the arrow polynomial R_G (Definition 4.6), and its proof relies on the correspondence H ↔ H∆F preserving the reduced boundary contributions. Theorems 5.7 and 5.8 are then dismissed as 'Analogous', but they require exactly the same partial-duality invariance for the twisted polynomial R^t_G (Definition 4.7) and the loop polynomial R^ℓ_G (Definition 4.8). For R^t_G, the boundary variables Ki, Λi are computed after applying the bar-arrow reduction rules of Figures 8 and 9. A bar is a point on a vertex boundary; partial duality re-pairs the vertex/edge arcs of the boundary, so a bar can move from one boundary component to another in the correspondence of Proposition 4.13. The proof of Proposition 4.13 does not track bars and gives no reason that the product of reduced boundary variables is preserved. For R^ℓ_G, the situation is worse: Definition 4.8 does not define the index ℓ; it must encode which boundary components separate the marked vertex from the puncture in the planar state, and the behaviour of this nesting data under partial duality is not discussed. Until the partial-duality identities for R^t_G and R^ℓ_G are proved, or the state-independence is proved directly, the right-hand sides of Theorems 5.7 and 5.8 may depend on the chosen state s, and the theorems are unsupported.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":1845,"tokens_out":1626,"duration_ms":69875,"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":[{"comment":"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.","section":"§5, Theorems 5.7 and 5.8; Proposition 4.13"},{"comment":"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.","section":"Proposition 2.10"},{"comment":"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.","section":"Lemmas 3.5 and 3.7"},{"comment":"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.","section":"Definition 6.6 and Theorem 6.8"}],"minor_comments":[{"comment":"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.","section":"Theorems 5.5, 5.7, 5.8"},{"comment":"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.","section":"Definition 4.8"},{"comment":"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}.","section":"Example 5.10"},{"comment":"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.","section":"Remark 3.16 and Remark 4.12"}],"recommendation":"major_revision","confidential_remarks":"The central state-sum idea is sound and the examples are convincing, but the manuscript repeatedly relies on 'analogous' proofs for statements that are not literally analogous: the twisted and loop Bollobás–Riordan polynomials have additional data whose behaviour under partial duality is not a special case of Proposition 4.13. I believe these gaps are repairable within the manuscript's framework, so I am not recommending rejection. The authors should also be asked to make the smoothing assertion in Proposition 2.10 rigorous, since it is the topological foundation for the twisted setting."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my read. The paper does something genuinely useful: it gives Thistlethwaite-type theorems for arrow polynomials of knotoids, which earlier work only gestured at, and it introduces twisted knotoids and marked ribbon graphs as the right framework. Theorem 5.5, the virtual knotoid case, is the centerpiece, and the proof is a genuine state-correspondence argument—not a restatement. The examples in Section 5 are concrete and checkable. The H-curve model for twisted knotoids is also a nice contribution, even if it needs tightening.\n\nThe soft spots are real and one of them is load-bearing. Proposition 4.13 proves partial-duality invariance only for the arrow polynomial RG. Theorems 5.7 and 5.8 are dismissed as 'Analogous', but their right-hand sides involve the twisted polynomial Rt and the loop polynomial Rl, and for those the partial-duality invariance is not proved anywhere. Since Theorems 5.7 and 5.8 assert equality with an expression that should be independent of the chosen state, and independence comes exactly from that missing invariance, this is not a cosmetic omission. In the twisted case, bars live on vertex boundaries and partial duality re-pairs boundary arcs, so it is not obvious the product of reduced boundary variables is preserved. In the loop case, Definition 4.8 does not even define the index ℓ on the c and Λ factors—it is just there in the formula. The reader's stress-test note lands on this correctly. The proof of Proposition 2.10 also contains a nontrivial 'it is easy to see' claim about keeping H-curves standard throughout an isotopy; that should be proved. Lemmas 3.5–3.7 are sketches, though they look fillable.\n\nNone of this makes the main idea wrong. The virtual theorem 5.5 appears solid, and the framework is worth having. But as written, the paper's advertised scope—twisted and planar, plus linkoids—is ahead of its proofs. I'd send it to a serious referee with a clear request: prove the partial-duality identities for Rt and Rl, or prove state-independence directly, and define ℓ. If those gaps close, it's a good paper; if they don't, the last two theorems need to be downgraded to conjectures.\n\nFor a reading group: maybe, if your group works on knotoids or Bollobás-Riordan polynomials. Would I cite it? Yes, for Theorem 5.5 and the marked ribbon graph construction. Would I accept for peer review? Yes—it deserves referee time, not a desk reject.","headline":"Real Thistlethwaite extension for knotoids with a solid virtual case, but the twisted and planar theorems depend on an unproved partial-duality invariance.","tokens_in":20657,"tokens_out":2519,"would_cite":true,"duration_ms":23406,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["57K10","57K12","05C31"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["knotoids","linkoids","twisted knotoids","arrow polynomial","loop arrow polynomial","Bollobás-Riordan polynomial","marked ribbon graphs","Thistlethwaite theorem"],"falsifier":"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.","tokens_in":19653,"feed_emoji":"🪢","tokens_out":9368,"duration_ms":81976,"temperature":0.7,"pith_summary":"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.","feed_headline":"Knotoid arrow polynomial is one ribbon-graph evaluation","feed_subtitle":"Classical Thistlethwaite theorem extends to open-ended knot diagrams on orientable and non-orientable surfaces","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the original Thistlethwaite theorem, the spanning-tree evaluation of the Jones polynomial that this paper generalizes to knotoids.","marker":"[25]"},{"why":"Introduces knotoids and the Turaev loop bracket, which are the primary objects and one of the target invariants.","marker":"[26]"},{"why":"Provides the theory of twisted links, stable equivalence, and the thickened-surface model that underlies twisted knotoids and H-curves.","marker":"[2]"},{"why":"Defines the Kauffman bracket state sum and the notion of Kauffman states on which the arrow polynomial and the ribbon graph construction depend.","marker":"[17]"},{"why":"First realizes the Kauffman bracket of virtual links as a Bollobás-Riordan polynomial evaluation, the template for the knotoid version.","marker":"[5]"},{"why":"Generalizes the virtual Thistlethwaite theorem to arbitrary Kauffman states via partial duality, giving the state-independence mechanism used here.","marker":"[4]"},{"why":"Develops arrow ribbon graphs and the arrow Bollobás-Riordan polynomial, including the deletion-contraction and partial-duality properties extended to marked ribbon graphs.","marker":"[3]"},{"why":"Introduces the arrow polynomial for knotoids and its normalization, the virtual-knotoid invariant that Theorem 5.5 recovers from a graph evaluation.","marker":"[15]"},{"why":"Defines the loop arrow polynomial for planar knotoids, the invariant behind the planar Thistlethwaite theorem of Theorem 5.8.","marker":"[14]"},{"why":"Supplies the theory of linkoids and the state-decomposition counterexamples that shape the half-integer arrow states and the linkoid theorem.","marker":"[11]"}],"fun_headline_variants":["Knotoid bracket becomes ribbon-graph polynomial","Thistlethwaite theorem generalized to knotoids","Arrow polynomial of knotoids via ribbon graphs","Linkoids and knotoids tie to Bollobás-Riordan","Extending Thistlethwaite to open-ended diagrams"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Knotoid bracket becomes ribbon-graph polynomial","Thistlethwaite theorem generalized to knotoids","Arrow polynomial of knotoids via ribbon graphs","Linkoids and knotoids tie to Bollobás-Riordan","Extending Thistlethwaite to open-ended diagrams"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00064,"raw_usage":{"total_tokens":2942,"prompt_tokens":939,"completion_tokens":2003,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":555,"completion_tokens_details":{"reasoning_tokens":1942}},"tokens_in":555,"tokens_out":2003,"duration_ms":14406,"temperature":1.0,"reasoning_tokens":1942,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T14:08:55.192260+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"A spanning tree expansion for the Jones polynomial","cited_arxiv_id":null,"evidence_quote":"Supplies the original Thistlethwaite theorem, the spanning-tree evaluation of the Jones polynomial that this paper generalizes to knotoids."},{"cited_title":"Knotoids","cited_arxiv_id":null,"evidence_quote":"Introduces knotoids and the Turaev loop bracket, which are the primary objects and one of the target invariants."},{"cited_title":"Twisted link theory","cited_arxiv_id":null,"evidence_quote":"Provides the theory of twisted links, stable equivalence, and the thickened-surface model that underlies twisted knotoids and H-curves."},{"cited_title":"State models and the Jones polynomial","cited_arxiv_id":null,"evidence_quote":"Defines the Kauffman bracket state sum and the notion of Kauffman states on which the arrow polynomial and the ribbon graph construction depend."},{"cited_title":"The Kauffman bracket of virtual links and the Bol- lob´ as-Riordan polynomial.Moscow Mathematical Journal , 7(3):409–418, 2007","cited_arxiv_id":null,"evidence_quote":"First realizes the Kauffman bracket of virtual links as a Bollobás-Riordan polynomial evaluation, the template for the knotoid version."},{"cited_title":"Generalized duality for graphs on surfaces and the signed Bollob´ as- Riordan polynomial","cited_arxiv_id":null,"evidence_quote":"Generalizes the virtual Thistlethwaite theorem to arbitrary Kauffman states via partial duality, giving the state-independence mechanism used here."},{"cited_title":"Arrow ribbon graphs","cited_arxiv_id":null,"evidence_quote":"Develops arrow ribbon graphs and the arrow Bollobás-Riordan polynomial, including the deletion-contraction and partial-duality properties extended to marked ribbon graphs."},{"cited_title":"New invariants of knotoids.European Journal of Combinatorics , 65:186–229, 2017","cited_arxiv_id":null,"evidence_quote":"Introduces the arrow polynomial for knotoids and its normalization, the virtual-knotoid invariant that Theorem 5.5 recovers from a graph evaluation."},{"cited_title":"Topological models for open-knotted protein chains using the concepts of knotoids and bonded knotoids","cited_arxiv_id":null,"evidence_quote":"Defines the loop arrow polynomial for planar knotoids, the invariant behind the planar Thistlethwaite theorem of Theorem 5.8."},{"cited_title":"Invariants of multi-linkoids.Mediter- ranean Journal of Mathematics , 20(3):165, 2023","cited_arxiv_id":null,"evidence_quote":"Supplies the theory of linkoids and the state-decomposition counterexamples that shape the half-integer arrow states and the linkoid theorem."}],"review_version":1}