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REVIEW 2 major objections 8 minor 32 references

Newton polygons for the non-bipartite dimer model

T0 review · 2 major / 8 minor · reviewed 2026-07-09 · glm-5.2

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Pith's one-line read Non-bipartite dimer Newton polygons equal their bipartite origins

desk verdict Solid extension of Newton polygon / zig-zag correspondence to non-bipartite dimer models; Theorem 39's general real-rootedness proof has a genuine gap read the letter →

arxiv 2607.07503 v1 pith:LR6TQOHP submitted 2026-07-08 math.CO math-phmath.MP

Vladimir Bošković This is my paper · ORCID
classification math.COmath-phmath.MP
keywords graphsnewtonbipartitedimerpolygonstorusgraphlattice
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 dimer model's algebraic geometry—Newton polygons, spectral curves, marginal polynomials—has been developed primarily for bipartite graphs, where a clean bipartite Kasteleyn matrix ensures all contributions to a given monomial share the same sign. This paper extends the theory to two families of non-bipartite graphs on the torus: graphs obtained by replacing degree-3 vertices of a bipartite graph with triangles (the blow-up operation, denoted Γ_ΔS), and corner graphs (C_Γ), where each vertex-face incidence becomes a vertex. The central discovery is that for isoradial bipartite graphs, the Newton polygon is invariant under the triangle blow-up: N(Γ_ΔS) = N(Γ) for any subset S of degree-3 vertices. For corner graphs, the polygon doubles: N(C_Γ) = 2N(Γ). In both cases, the primitive edge vectors of the Newton polygon are exactly the homology classes of the zig-zag paths. The paper then proves that the marginal polynomials (restrictions to boundary sides of the polygon) are real-rooted for these non-bipartite families, with explicit factorizations for the triangular lattice and the hexagonal Fisher graph. Finally, the paper introduces new local moves—the λ-move, diagonal spider move, and a non-planar cross move—that preserve the dimer partition function up to scale, extending the equivalence toolkit beyond the bipartite planar setting.

What carries the argument

isoradial graph; zig-zag path; Newton polygon; characteristic polynomial; Kasteleyn matrix; triangle blow-up (Γ_Δ); corner graph (C_Γ); marginal polynomial; local moves (λ-move, diagonal spider move, cross move)

What would settle it

A non-isoradial bipartite graph where replacing a degree-3 vertex with a triangle changes the Newton polygon (the author provides one in Remark 23), or a non-bipartite isoradial graph whose marginal polynomial has a non-real root.

Watch

Extended reading notes

Core claim

The Newton polygon of a non-bipartite dimer model can be identical to that of its underlying bipartite graph, provided the graph is isoradial. The triangle blow-up operation Γ → Γ_ΔS preserves the polygon because zig-zag path homology classes are in bijection between the two graphs (Proposition 12), and the isoradiality condition ensures that boundary-crossing counts—which bound the polygon—are preserved (Lemma 15). For corner graphs, the polygon scales by a factor of 2 because each zig-zag path in Γ gives rise to two zig-zag paths in C_Γ with the same homology (Proposition 26). The marginal polynomials are real-rooted because the loop structure on boundary sides of the polygon forces a comb

Load-bearing premise

The isoradiality assumption on the underlying bipartite graph is load-bearing: it ensures zig-zag paths intersect at most once in the universal cover, which is used to bound boundary crossings and prove the polygon inclusion N(Γ_ΔS) ⊆ N(Γ). The author explicitly notes that this inclusion can fail for non-isoradial graphs, exhibiting a counterexample.

Editorial extensions

If this is right

  • The equality N(Γ_ΔS) = N(Γ) means the phase diagram (smooth, frozen, rough regions) of the non-bipartite dimer model on Γ_ΔS is the same as that of the bipartite model on Γ, extending limit-shape analysis to non-bipartite settings.
  • Real-rootedness of marginal polynomials for non-bipartite graphs is a necessary condition for the spectral curves to be simple Harnack curves; this paper establishes that condition for these families, partially answering the open question of spectral curve classification in the non-bipartite case.
  • The new local moves (λ-move, diagonal spider move, cross move) enlarge the equivalence classes of graphs sharing a characteristic polynomial, meaning results about Newton polygons and partition functions transfer to a broader family of non-bipartite and even non-planar graphs.
  • The cross move's connection to the cube recurrence and Kuo condensation suggests a route to enumerating perfect matchings of non-planar analogs of the Aztec diamond.
  • The intermediate containment N(Γ) ⊆ N(F_Γ) ⊆ 2N(Γ) for Fisher graphs suggests a graded structure where the Newton polygon interpolates between the original and doubled polygon, depending on the decoration.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 8 minor

Summary. This paper studies Newton polygons for the dimer model on two families of non-bipartite torus graphs: (1) graphs obtained by replacing degree-3 vertices of a bipartite isoradial graph with triangles (Γ_ΔS), and (2) corner graphs (C_Γ). The main results (Theorems 1/22 and 2) establish that N(Γ_ΔS) = N(Γ) for isoradial bipartite Γ, and N(C_Γ) = 2N(Γ), with primitive edge vectors given by zig-zag path homology classes. The proofs proceed by double inclusion: the forward inclusion (Lemma 16) uses a counting argument (Lemma 15) relying on isoradiality, while the reverse inclusion (Lemma 21) uses realizability of extremal homology classes (Lemmas 19, 20). Section 5 addresses real-rootedness of marginal polynomials for the triangular lattice (Proposition 35), the hexagonal Fisher graph (Theorem 38), and a general statement for isoradial bipartite graphs (Theorem 39). Section 6 introduces new local moves (λ-move, diagonal spider move, cross move) preserving dimer partition functions and characteristic polynomials.

Significance. The paper extends the Newton polygon / zig-zag path correspondence from the well-studied bipartite dimer model to two natural non-bipartite families. The triangle-replacement construction Γ_ΔS is directly motivated by the Fisher/Ising correspondence, making the results relevant to understanding spectral curves of non-bipartite dimer models. The explicit factorizations of marginal polynomial roots for the triangular and hexagonal Fisher lattices (Proposition 35, Theorem 38) are concrete and verifiable. The new local moves in Section 6, including the non-planar cross move and its connection to the cube recurrence, are a genuine addition to the combinatorial toolkit. The double-inclusion proof strategy is clean and the isoradiality assumption is properly identified as load-bearing (Remark 23).

major comments (2)
  1. Theorem 39 (Section 5): The proof claims that 'the only two ways in which these paths can be connected are u_{i,1}→u_{i,2},…,u_{i,k-1}→u_{i,k} or u_{i,1}→u_{i,k},…,u_{i,k-1}→u_{i,k-2}.' This is justified by a diagrammatic argument (Figure 16, right) showing one forbidden configuration, and the author notes this is 'the same argument we had in Lemma 37(c).' However, Lemma 37 is specific to the Fisher graph of the hexagonal lattice, where the local structure is rigid. For a general isoradial bipartite graph, the neighborhood of each vertex in Z_i can be arbitrary (subject to isoradiality), and the claim that exactly two path configurations exist is not formally proven. If a third configuration exists for some isoradial graph, the factor P_{U_i^-→U_i^+}(w) could have degree > 1, and real-rootedness would not follow. This is load-bearing for Theorem 39's central claim. The author should要么 (a
  2. Lemma 15 (Section 3): The proof argues that if a zig-zag path Z has homology (0,d) with d≠0, it would intersect Z_1 more than once, contradicting isoradiality. However, the argument that 'such a path would necessarily intersect Z_1 more than once within the fundamental domain' needs more justification. A path with homology (0,d) is homologous to a vertical path, but it could potentially wind around the torus in a way that avoids Z_1 except at shared edges. The isoradiality condition forbids two distinct zig-zag paths from intersecting more than once in the universal cover, but the argument should explicitly connect the homology class (0,d) to the geometric necessity of multiple intersections with Z_1. This step is load-bearing for Lemma 15, which is in turn load-bearing for Lemma 16 and Theorem 22.
minor comments (8)
  1. Section 2.2, Definition 9: The definition of isoradiality for general (non-bipartite) graphs is given as 'no self-intersecting zig-zag paths and no two zig-zag paths intersect more than once.' This is sometimes called 'minimal' in the bipartite literature. The relationship between the non-bipartite isoradiality defined here and the standard rhombic embedding / isoradial embedding notion should be clarified.
  2. Theorem 1 vs Theorem 22: Theorem 1 in the introduction states the result for 'isoradial bipartite torus graph,' while Theorem 22 in Section 3 uses 'isoradial bipartite graph.' These should be consistent. Also, Theorem 1 adds the claim about primitive edge vectors for 3-valent graphs, which appears as Corollary 24 in the body. The relationship should be stated explicitly.
  3. Section 4.3 (Fisher graphs): The results N(Γ)⊆N(F_Γ)⊆2N(Γ) are stated without proof ('we omit the proofs of these results'). While the author notes reliance on previous techniques, for a journal publication these proofs should be included, or the results should be clearly labeled as conjectures/observations rather than propositions (Proposition 34).
  4. Figure 16: The left and right panels use color (blue, red, green) to distinguish edge types and paths. For print accessibility, consider adding labels or patterns so the figure can be parsed without color.
  5. Section 5, notation: The polynomials P_R, P_U are introduced in Section 5.1 for the triangular lattice and reused in Section 5.2 for the Fisher graph with potentially different meanings. Clarifying the scope of notation would help.
  6. Proposition 42 (Section 6.1): The case analysis is thorough but lengthy. A summary table mapping each case to its contribution would improve readability.
  7. Section 6.2: The connection to the cube recurrence and Kuo condensation is mentioned briefly. A reference to the specific form of Kuo condensation being invoked would strengthen this remark.
  8. References: Reference [7] (de Tilière and Rey) is dated 2026 with arXiv number 2602.13526. This appears to be a forward-dated preprint. The author should verify this reference is accessible and correctly cited.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for a careful reading and for identifying two points in the proofs that require additional justification. Both points concern gaps in the argumentation that we will address in revision: one in the proof of Theorem 39 (general isoradial case) and one in the proof of Lemma 15 (intersection count). We agree with the referee that both are load-bearing and will revise accordingly.

read point-by-point responses

  1. Referee: Theorem 39 (Section 5): The proof claims that 'the only two ways in which these paths can be connected are u_{i,1}→u_{i,2},…,u_{i,k-1}→u_{i,k} or u_{i,1}→u_{i,k},…,u_{i,k-1}→u_{i,k-2}.' This is justified by a diagrammatic argument (Figure 16, right) showing one forbidden configuration, and the author notes this is 'the same argument we had in Lemma 37(c).' However, Lemma 37 is specific to the Fisher graph of the hexagonal lattice, where the local structure is rigid. For a general isoradial bipartite graph, the neighborhood of each vertex in Z_i can be arbitrary (subject to isoradiality), and the claim that exactly two path configurations exist is not formally proven. If a third configuration exists for some isoradial graph, the factor P_{U_i^-→U_i^+}(w) could have degree > 1, and real-rootedness would not follow. This is load-bearing for Theorem 39's central claim. The author should (a)

    Authors: The referee is correct that the argument as written does not adequately justify the claim for general isoradial bipartite graphs. The reference to Lemma 37(c) is insufficient because that lemma exploits the rigid local structure of the hexagonal Fisher graph, which does not transfer to the general isoradial setting. We have identified the key missing ingredient: the argument needs the fact that the curve γ_x intersects each zig-zag path Z_i exactly once, and that the triangles replacing vertices of S are sufficiently small so that γ_x crosses at most one edge in the corresponding portion of Γ_{ΔS}. This geometric constraint, combined with the requirement that the paths from U_i^- to U_i^+ must be disjoint and realizable within L(Γ_{ΔS}), is what forces the degree of P_{U_i^-→U_i^+}(w) to be at most one. The referee's concern about a potential third configuration is well-taken: in the current draft, we only exhibit one forbidden configuration (Figure 16, right) and do not systematically rule out all others. In the revision, we will provide a complete case analysis showing that any configuration other than the two stated would either (i) force a path to cross γ_x more than once, contradicting the degree-one bound, or (ii) create a vertex of degree 3 in the collection of loops, which is forbidden. We will also add a remark clarifying that the isoradiality condition enters through the edge-disjointness of zig-zag paths with the same homology class, which ensures the choices for different Z_i are independent. If, upon completing this analysis, we find that the general case cannot be fully justified, we will restrict Theorem 39 to the families where the proof is complete (triangular lattice, hexagonal Fisher graph, and graphs where all vertices of Z_i lie in S) and state the revision: no

  2. Referee: Lemma 15 (Section 3): The proof argues that if a zig-zag path Z has homology (0,d) with d≠0, it would intersect Z_1 more than once, contradicting isoradiality. However, the argument that 'such a path would necessarily intersect Z_1 more than once within the fundamental domain' needs more justification. A path with homology (0,d) is homologous to a vertical path, but it could potentially wind around the torus in a way that avoids Z_1 except at shared edges. The isoradiality condition forbids two distinct zig-zag paths from intersecting more than once in the universal cover, but the argument should explicitly connect the homology class (0,d) to the geometric necessity of multiple intersections with Z_1. This step is load-bearing for Lemma 15, which is in turn load-bearing for Lemma 16 and Theorem 22.

    Authors: The referee identifies a genuine gap in the proof of Lemma 15. The claim that a zig-zag path with homology (0,d), d≠0, must intersect Z_1 more than once is stated without sufficient justification. The key observation is that after applying Lemma 14, the fundamental domain is chosen so that the vertical boundary crosses every edge of Z_1 and no other edges of Γ. A zig-zag path Z with homology (0,d) is homologous to d times the vertical generator, meaning it must cross the vertical boundary of the fundamental domain at least |d| times (with appropriate signs). Since the only edges crossing the vertical boundary are edges of Z_1, the path Z must share at least |d| edges with Z_1 in the fundamental domain. When we lift to the universal cover, each such shared edge gives rise to a distinct intersection between a lift of Z and a lift of Z_1. If |d| ≥ 2, this yields at least two intersections in the universal cover, contradicting isoradiality. If |d| = 1, then Z is homologous to Z_1, and one must argue that two distinct zig-zag paths with the same homology class (0,1) that share an edge must in fact coincide (since they both turn maximally left and right at each vertex), which would make them the same path, not two distinct paths. We agree that this chain of reasoning is not spelled out in the current draft and will add it explicitly in the revision. revision: no

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity found; derivation chain is self-contained

full rationale

The paper's main results (Theorems 1/22, 2, 3/39, Proposition 35, Theorem 38) are derived from standard definitions of the Kasteleyn matrix, Newton polygons, zig-zag paths, and loop configurations, combined with externally verifiable prior results (Lemma 4 from Boutillier-de Tilière [1], Lemma 8 and Theorem 5 from Goncharov-Kenyon [15]). No result is defined in terms of its own conclusion, no parameter is fitted and then presented as a prediction, and no self-citation chain forces the conclusion. The proof of Theorem 22 proceeds by genuine double inclusion (Lemma 16 ⊆, Lemma 21 ⊇), each using independent combinatorial arguments. Theorem 39's proof uses Theorem 22 to constrain the degree of the factored polynomial — this is standard mathematical reasoning (using a previously proven result as input), not circularity. The explicit root formulas in Proposition 35 and Theorem 38 are direct computations from the loop structure, not fits. The one minor self-citation is to [1] (Lemma 4), where the author's PhD advisor is a co-author; however, this lemma is a standard, independently verifiable result about loop decompositions of the characteristic polynomial, widely used in the dimer literature, and is not load-bearing in a circular sense. The skeptic's concern about Theorem 39's 'only two ways' argument being insufficiently general is a correctness gap, not a circularity issue. Score 1 reflects the minor self-citation that is not load-bearing for the central claims in a circular way.

Assumptions & free parameters 0 free parameters · 5 assumptions · 3 invented entities

No free parameters are fitted to data. The axioms are standard results from the dimer model literature plus the isoradiality domain assumption. The invented entities are local moves, each with an independent combinatorial proof of partition function preservation.

assumptions (5)
  • standard math The Kasteleyn matrix determinant computes the dimer partition function for planar graphs (Kasteleyn [16], Temperley-Fisher [31]).
    Used in Section 2.1 as the foundational tool for computing partition functions.
  • standard math Lemma 4 from Boutillier-de Tilière [1]: characteristic polynomial properties including loop expansion, odd trivial loop cancellation, parallel homology of non-trivial loops, and central symmetry P(z,w)=P(z^{-1},w^{-1}).
    Invoked throughout Sections 2-5 as the combinatorial foundation for analyzing characteristic polynomials of non-bipartite graphs.
  • standard math Lemma 8 from Goncharov-Kenyon [15]: bijection between bipartite zig-zag path homology classes and primitive edge vectors of Newton polygons for minimal bipartite graphs.
    Used in Section 2.2 and Corollary 10 to establish the bipartite baseline that the paper extends to non-bipartite settings.
  • domain assumption Isoradiality of the underlying bipartite graph: zig-zag paths do not self-intersect and pairwise intersect at most once in the universal cover.
    Required for Lemma 15 (counting boundary crossings) and Lemma 16 (forward inclusion of Newton polygon equality). The author notes in Remark 23 that the result can fail without this assumption.
  • ad hoc to paper Generic edge weights for Fisher graphs in Proposition 34.
    Proposition 34 explicitly assumes generic edge weights, noting this is not the case for the standard Fisher correspondence where decoration edges have weight 1.
invented entities (3)
  • λ-move (Figure 17) independent evidence
    purpose: A local move on non-bipartite graphs with a degree-3 internal vertex that preserves the dimer partition function.
    Lemma 40 provides a direct combinatorial proof that the partition function is preserved, verified by matching dimer cover contributions.
  • Diagonal spider move (Figure 18) independent evidence
    purpose: Extension of the standard spider move to non-bipartite graphs via an added diagonal edge, preserving partition function up to scale 1/Δ.
    Lemma 41 proves preservation by reducing to the standard spider move and checking the diagonal edge case.
  • Cross move (Figure 19) independent evidence
    purpose: A local move that can produce non-planar graphs, preserving the partition function up to scale 1/Δ* where Δ*=ac+bd+ef.
    Lemma 43 proves preservation by case analysis on which edges appear in dimer covers. The connection to the cube recurrence is noted as a suggestion, not proven.

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Pith. "Pith review of Newton polygons for the non-bipartite dimer model." pith.science (2026). https://pith.science/paper/LR6TQOHP

@misc{pith2026260707503,
  author       = {Pith},
  title        = {Pith review of: Newton polygons for the non-bipartite dimer model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LR6TQOHP}},
  note         = {Machine review of arXiv:2607.07503}
}
abstract

We study the dimer model on two families of non-bipartite graphs on a torus. The first family is obtained by replacing degree $3$ vertices in a bipartite torus graph with triangles, while the second consists of corner graphs associated with bipartite torus graphs. We determine the relationship between the Newton polygons of these graphs and those of the underlying bipartite graphs. We also identify the primitive edge vectors of the Newton polygons with the homology classes of the zig-zag paths. We further consider the marginal polynomials obtained by restricting to monomials corresponding to a boundary side of the Newton polygon. For the triangular lattice and the Fisher graph of the hexagonal lattice, we prove that these polynomials are real-rooted and obtain an explicit factorization of their roots. Finally, we introduce new local moves, including a move on non-planar graphs, that preserve the dimer partition functions up to a scale.

Figures

Figures reproduced from arXiv: 2607.07503 by the authors.

Figure 1
Figure 1. From left to right: the hexagonal lattice [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Fisher graph F1,1 with its Kasteleyn orientation, positive real edge weights a, b, c and the remaining edges have weight 1, and the Newton polygon for its characteristic polynomial. We modify the definition of the Kasteleyn matrix to make it depend on the variables z, w ∈ C ∗ . For each edge e connecting vertices u and v in Γ, there are three contributions: its weight wt(e), its Kasteleyn sign κ(e), and a monomial χ… view at source ↗
Figure 3
Figure 3. A spider move and an expansion/contraction of a degree [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (17 more)
Figure 4
Figure 4. Figure 4: Hexagonal lattice, its zig-zag paths, and the corresponding Newton [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: The correspondence between the zig zag paths of [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 6
Figure 6. Figure 6: Left: a torus graph Γ, a zig-zag path Z with homology class (0, 1), and a loop γZ. Right: the same graph Γ obtained by deforming γZ into a straight curve, and moving appropriately the vertices. This operation does not change the homology classes of any zig-zag path in …
Figure 7
Figure 7. Figure 7: The correspondence between the loops of Γ and Γ∆v . In the second case, v is adjacent to two different vertices in γ, say u1 and u2. Then we can replace the path u1vu2 of the loop in γ with the path u1v1v3v2u2. See [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]
Figure 8
Figure 8. Figure 8: The graph on the left contains no collection of loops with total homology [PITH_FULL_IMAGE:figures/full_fig_p018_8.png]
Figure 9
Figure 9. Figure 9: A zig-zag path Z in Γ gives rise to two zig-zag paths Z ′ and Z ′′ in the corner graph CΓ. Lemma 27. Let G be a connected graph. Any spanning forest F of G can be extended to a spanning tree of G. Proof. Suppose F consists of k ≥ 2 disjoint trees (if k = 1, we are done…
Figure 10
Figure 10. Figure 10: Up: a collection of loops γ in blue with homology (a, b), extended with red edges into a cycle-rooted spanning forest. Down: the corresponding collection of loops γc without double edges and homology (2a, 2b). 1This is the same map as defined earlier in the section. 2…
Figure 11
Figure 11. Figure 11: The zig-zag paths on the graph Γ, and the corresponding zig-zag paths in FΓ, containing an extra zig-zag path of z ′ 0 of homology zero. For these graphs, we also discuss their Newton polygons and zig-zag paths. We observe an intermediate phenomenon where the Newton p…
Figure 12
Figure 12. Figure 12: The square lattice S, the graph FS and the comparison of their Newton polygons. 5 Marginal polynomials In this section, we study the roots of the polynomials that correspond to the boundary sides of the Newton polygons, which we call marginal polynomials. For minimal …
Figure 13
Figure 13. Figure 13: Triangular lattice T2,2 and its Newton polygon. While all the c-edges carry the same Kasteleyn orientation, the orientation of the a-edges and of the b-edges depends on the parity of their first coordinate: to the right if it is odd, and to the left if it is even. We …
Figure 14
Figure 14. Figure 14: The Fisher graph F2,3 and its Newton polygon. Proof. (a) We start by partitioning the set of b-edges into {B1, . . . , Bn}, where Bk = {b1,k, . . . , bm,k} for all 1 ≤ k ≤ n. In order to obtain a monomial that contains z m, we need all the edges of Bn in the collectio…
Figure 15
Figure 15. Figure 15: An operation that resolves double edges in the collections of loops. [PITH_FULL_IMAGE:figures/full_fig_p029_15.png]
Figure 16
Figure 16. Figure 16: Left: the portion of the graph Γ containing the zig-zag path Zi with homology (0, 1), its a-edges are depicted in blue, and b-edges are in red. The curve γx is chosen such that it intersect every zig-zag path Zi exactly once. Right: the portion of the graph Γ∆S obtain…
Figure 16
Figure 16. Figure 16: In order to describe the collections of loops in Γ∆S whose first coordinate of the homology class is equal to k/2, instead of choosing a-edges and b-edges in Zi , we need to consider the disjoint oriented paths going from U − i = {ui,1, ui,3, . . . , ui,k−1} to U + i …
Figure 17
Figure 17. Figure 17: , we define the λ-move, which involves four vertices such that its internal vertex has degree three, and the remaining vertices can be connected to the rest of the graph (which can also include an edge between the vertices v2 and v3). a b c d b c d ad b v1 v2 v3 u v1 …
Figure 18
Figure 18. Figure 18: A diagonal spider move, ∆ := ac + bd. whether the dimer covers that contain the edges with respective weights e and e ∆ have the same total weights up to a factor of ∆. If e is in a dimer cover, then the remaining four vertices in G of [PITH_FULL_IMAGE:figures/full_f…
Figure 19
Figure 19. Figure 19: A cross move, ∆∗ := ac + bd + ef. Lemma 43. Let G′ be the graph obtained by applying a cross move on a graph G. Then Z(G ′ ) = 1 ∆∗ Z(G), where ∆∗ = ac + bd + ef. Proof. If we take e = f = 0, then we obtain the usual spider move. So it suffices to check the cases when…

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Pith tools

Reviewed July 9, 2026 · model on record in the stance chip above.