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arxiv: 2604.21942 · v1 · submitted 2026-04-15 · 🧮 math.CO

Partial Petrial Polynomials of Ribbon Graphs

Xiaoxiang Yu , Rong-Xia Hao , Jianbing Liu , Zhiguo Li This is my paper

Pith reviewed 2026-05-10 13:24 UTC · model grok-4.3

classification 🧮 math.CO
keywords ribbon graphpartial Petrial polynomialbouquetfour-term relation4-invariantsigned graphEuler genusgraph polynomial
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The pith

The partial Petrial polynomial of a ribbon graph with n vertices equals the sum of the same polynomial over 2^{n-1} distinct bouquets, and its signed modification is a 4-invariant.

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

The paper proves that the partial Petrial polynomial of any ribbon graph can be expressed as the sum of the polynomial over 2 to the power n minus one carefully chosen bouquets. This representation generalizes prior results limited to bouquets alone. The authors introduce a modified partial Petrial polynomial by multiplying selected terms by plus or minus one. They show this modification ensures the four-term relation holds and extend the construction to signed simple graphs, where the polynomial is invariant under that relation.

Core claim

The partial Petrial polynomial of a ribbon graph G with n vertices is equal to the sum of this polynomial for 2^{n-1} distinct bouquets. By assigning coefficients +1 or -1 to the terms in the partial Petrial polynomial, the resulting polynomial satisfies the four-term relation for graphs. This modified polynomial generalizes from bouquets to all signed simple graphs while remaining 4-invariant.

What carries the argument

The partial Petrial polynomial together with its modification by sign coefficients to enforce the four-term relation.

If this is right

  • The matrix rank characterization of Euler genus holds for all ribbon graphs, not just bouquets.
  • Polynomial computations for general ribbon graphs simplify to the bouquet case.
  • The modified polynomial serves as a 4-invariant for signed simple graphs.
  • It resolves the question of identifying 4-invariants among known graph polynomials.

Where Pith is reading between the lines

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

  • The bouquet sum representation allows properties of the polynomial to be transferred from simple cases to general ones.
  • Similar reductions to bouquets could apply to other polynomials defined on ribbon graphs.
  • The 4-invariant may be used to distinguish signed graphs up to four-term equivalence.

Load-bearing premise

Assigning coefficients of +1 or -1 to terms in the partial Petrial polynomial creates a version that obeys the four-term relation on all ribbon graphs and extends directly to signed simple graphs.

What would settle it

A single ribbon graph whose partial Petrial polynomial value does not match the sum of the polynomial values on 2^{n-1} bouquets, or a signed graph where the modified polynomial fails to stay constant under four-term moves.

Figures

Figures reproduced from arXiv: 2604.21942 by Jianbing Liu, Rong-Xia Hao, Xiaoxiang Yu, Zhiguo Li.

Figure 1
Figure 1. Figure 1: Four-term relation for framed chord diagrams. [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The ribbon graph G/e, where the edge e in (a) (resp. (b)) is untwisted (resp. twisted). Lemma 3.1. Let G be a connected ribbon graph. Assume that T is a spanning tree of G. Then ε(G) = ε(Aux(G, T)). Proof. Recall that the ribbon graph Aux(G, T) is obtained from G by contracting all edge-ribbons in T. Since ribbon graphs are surfaces with boundaries, the surface Aux(G, T) can be seen as being obtained from … view at source ↗
Figure 3
Figure 3. Figure 3: Three kinds of transformations of the chord diagram [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The partial Petrial graphs B×|A, Bea,b ×|A, Be′ a,b ×|A, and B′ a,b ×|A, where two bouquets in boxes of the same color can be obtained by sliding one edge. 10 [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
read the original abstract

Gross, Mansour, and Tucker [European J. Combin., 95 (2021): 103329] introduced the \emph{partial Petrial polynomial} of a ribbon graph $G$, denoted by $^{\partial}{\varepsilon^{\times}_{G}}(z)$. Beck and Mellor proved, in both orientable and non-orientable cases respectively, that the Euler genus of a bouquet equals the rank of a certain matrix over $\mathbb{GF}(2)$. In this paper, we first generalize Beck and Mellor's results from bouquets to all ribbon graphs. Secondly, we give an equivalent representation of the partial Petrial polynomial for all ribbon graphs. Specifically, the partial Petrial polynomial of a ribbon graph $G$ with $n$ vertices is equal to the sum of this polynomial for $2^{n-1}$ distinct bouquets. Moreover, we give the definition of a modified partial Petrial polynomial by assigning coefficients $+1$ or $-1$ to the terms in the partial Petrial polynomial such that the resulting polynomial satisfies the four-term relation for graphs. Finally, we generalize the modified partial Petrial polynomial from bouquets to all signed simple graphs and prove that this polynomial is $4$-invariant, which provides an answer to the problem posed by Lando [J.~Combin.~Theory Ser.~B,~80~(1) (2000): 104-121]: Which of the known graph invariants are $4$-invariants?

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper generalizes Beck and Mellor's matrix-rank characterization of Euler genus from bouquets to arbitrary ribbon graphs. It proves that the partial Petrial polynomial of an n-vertex ribbon graph equals the sum of the same polynomial evaluated on 2^{n-1} distinct bouquets. A modified partial Petrial polynomial is then defined by assigning coefficients +1 or -1 to its terms so that the four-term relation holds; this modified polynomial is extended to signed simple graphs and shown to be 4-invariant, answering Lando's question on 4-invariants.

Significance. If the bouquet decomposition and the 4-invariance proof hold, the work is significant: it supplies a new 4-invariant for signed graphs and resolves an open problem of Lando. The structural reduction to bouquets is a concrete computational and theoretical tool that could be used in further studies of ribbon-graph polynomials.

major comments (2)
  1. [section introducing the modified partial Petrial polynomial] Definition of the modified partial Petrial polynomial: the signs are introduced by the phrase 'assigning coefficients +1 or -1 ... such that the resulting polynomial satisfies the four-term relation.' For the subsequent claim that the modified polynomial on signed simple graphs is 4-invariant to follow from the bouquet case via the 2^{n-1}-sum, the sign rule must be shown to be intrinsic (e.g., a fixed function of the edge signs or vertex data) rather than chosen per graph. Without an explicit canonical construction, the transfer of 4-invariance across the bouquet decomposition is not guaranteed.
  2. [section generalizing Beck-Mellor results] Generalization of Beck-Mellor: the claim that Euler genus equals the GF(2)-rank of a certain matrix is extended from bouquets to all ribbon graphs. The matrix is not defined for graphs with more than one vertex in the visible text; the proof that this rank equals the Euler genus must be checked for consistency with the partial Petrial polynomial before the bouquet-sum representation can be used.
minor comments (1)
  1. The notation ^{(∂)}ε^×_G(z) is introduced without a reminder of its precise relation to the ordinary Petrial polynomial; a one-sentence comparison would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the valuable comments provided. We respond to each major comment as follows, and will incorporate revisions where indicated.

read point-by-point responses

  1. Referee: [section introducing the modified partial Petrial polynomial] Definition of the modified partial Petrial polynomial: the signs are introduced by the phrase 'assigning coefficients +1 or -1 ... such that the resulting polynomial satisfies the four-term relation.' For the subsequent claim that the modified polynomial on signed simple graphs is 4-invariant to follow from the bouquet case via the 2^{n-1}-sum, the sign rule must be shown to be intrinsic (e.g., a fixed function of the edge signs or vertex data) rather than chosen per graph. Without an explicit canonical construction, the transfer of 4-invariance across the bouquet decomposition is not guaranteed.

    Authors: We agree with the referee that the sign assignment requires an explicit and intrinsic definition to ensure the 4-invariance transfers correctly through the bouquet decomposition. The manuscript defines the modified polynomial such that the four-term relation is satisfied, but to address this concern, we will revise the section to include a canonical construction. Specifically, we will define the signs as a fixed function of the edge signs in the signed graph, chosen consistently (for example, by the product of signs in certain cycles or a parity-based rule derived from the bouquet representation). This ensures the rule is independent of the particular graph and allows the sum over bouquets to preserve the property. We believe this strengthens the presentation. revision: yes

  2. Referee: [section generalizing Beck-Mellor results] Generalization of Beck-Mellor: the claim that Euler genus equals the GF(2)-rank of a certain matrix is extended from bouquets to all ribbon graphs. The matrix is not defined for graphs with more than one vertex in the visible text; the proof that this rank equals the Euler genus must be checked for consistency with the partial Petrial polynomial before the bouquet-sum representation can be used.

    Authors: The generalization of the Beck-Mellor matrix-rank characterization is presented in the manuscript by extending the definition from bouquets using the structure of ribbon graphs with multiple vertices. However, we acknowledge that the explicit matrix definition for n > 1 may not be sufficiently highlighted in the text. In the revision, we will provide the full definition of the matrix for arbitrary ribbon graphs and include a detailed verification that its GF(2)-rank equals the Euler genus, ensuring consistency with the partial Petrial polynomial. This consistency is established prior to invoking the bouquet-sum representation, as the polynomial is defined in terms of the same combinatorial data. We will add this clarification to make the proof self-contained. revision: yes

Circularity Check

1 steps flagged

Modified partial Petrial polynomial defined via sign choice to enforce four-term relation

specific steps
  1. self definitional [Abstract]
    "we give the definition of a modified partial Petrial polynomial by assigning coefficients +1 or -1 to the terms in the partial Petrial polynomial such that the resulting polynomial satisfies the four-term relation for graphs. Finally, we generalize the modified partial Petrial polynomial from bouquets to all signed simple graphs and prove that this polynomial is 4-invariant"

    The modified polynomial is introduced by choosing signs explicitly so that the four-term relation holds. The claim that the version extended to all signed simple graphs is 4-invariant therefore follows directly from the definitional choice of coefficients rather than from a separate, graph-intrinsic rule that independently produces the invariance.

full rationale

The paper establishes an equivalent sum representation over bouquets and generalizes prior Beck-Mellor results without evident circularity. However, the load-bearing step for the 4-invariance result on signed graphs is the definition of the modified polynomial, which selects coefficients specifically to satisfy the target relation. The subsequent generalization and proof then reduce to this construction rather than an independent, uniform assignment rule that would guarantee the property for arbitrary signed graphs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on standard combinatorial definitions of ribbon graphs, partial Petrial polynomials, and the four-term relation; no new free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (2)
  • standard math Standard algebraic properties of polynomials and the four-term relation for graphs
    Invoked when defining the modified polynomial and proving invariance.
  • domain assumption Properties of Euler genus and matrix rank over GF(2) for bouquets extend to general ribbon graphs
    Central to the first generalization step.

pith-pipeline@v0.9.0 · 5563 in / 1415 out tokens · 30171 ms · 2026-05-10T13:24:30.264975+00:00 · methodology

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Reference graph

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