Tensor product formulas for the Bollob\'as-Riordan and Krushkal polynomials

Iain Moffatt; Maya Thompson

REVIEW 4 major objections 4 minor 23 references

Tensor product formulas for the Bollob\'as-Riordan and Krushkal polynomials

T0 review · 4 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper proves a Brylawski-style tensor product formula that covers both the Krushkal and Bollobás-Riordan polynomials, unifying previously partial or special-case results.

desk verdict A genuinely new Krushkal tensor product formula, built on a useful packaged-arrow-presentation framework, but with several load-bearing lemmas deferred to 'routine' checks. read the letter →

arxiv 2505.22570 v1 pith:OQKRHF5F submitted 2025-05-28 math.CO

Iain Moffatt , Maya Thompson This is my paper

classification math.CO MSC 05C3105C10

keywords tensorproductformulaBollobás-RiordanpolynomialKrushkalpackagedarrowpresentationsembeddedgraphstopologicalTuttedeletion-contractionribbon

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

Brylawski's tensor product formula expresses the Tutte polynomial of a graph obtained by gluing a copy of a second graph onto each edge in terms of the two factors. This paper extends that formula to two harder topological analogues, the Krushkal and Bollobás-Riordan polynomials of graphs embedded in surfaces, which previously had only partial or special-case tensor product formulas. The key move is to work with packaged arrow presentations, a combinatorial model of graphs embedded in pseudo-surfaces that carries extra vertex and boundary partitions and supports five edge operations rather than two. The main theorem shows that a five-term transition polynomial $Q$ of a tensor product equals $Q$ of the first factor with each edge's parameters replaced by the solution of a $5\times5$ linear system built only from the second factor. From this single identity, Brylawski-style formulas for the Krushkal and Bollobás-Riordan polynomials follow, and known formulas for transition, ribbon graph, and Tutte polynomials are recovered as special cases.

What carries the argument

The machinery has three parts. Packaged arrow presentations are the objects: an arrow presentation (circles carrying arrows paired into edges) together with a partition of the vertex set and a partition of the boundary components; they model graphs non-cellularly embedded in pseudo-surfaces. On these objects the paper defines five edge operations, namely deletion, contraction, Penrose-contraction, merge-deletion, and merge-contraction, and a five-term polynomial $Q$ whose recursion applies any one of them to an arbitrary edge, with base case $\alpha^{|V(G)|}\beta^{|V|}\gamma^{|B|}$ on edgeless presentations. Two lemmas carry the argument: Lemma 2.3 says the five operations commute on distinct edges, and Lemma 2.6 says each operation is exactly a 2-sum with one of five one-edge packaged arrow presentations $K_1,\dots,K_5$. Theorem 3.6 combines these facts: resolving the second factor's edges in $Q$ turns each summed edge of the first factor into the solution of a $5\times5$ linear system whose coefficients encode the $Q$-values of those five one-edge reductions.

What would settle it

The most direct check is to compute both sides of Theorem 3.6 for a one-edge first factor and a non-trivial one-edge second factor with non-trivial vertex and boundary partitions; any mismatch in the resulting polynomials, or a failure of two of the five operations to commute on a two-edge packaged arrow presentation, would falsify the central claim.

Watch

Extended reading notes

Core claim

The central claim is Theorem 3.6: if $G[k]$ is a packaged arrow presentation obtained from $G$ by forming 2-sums with packaged arrow presentations $H^{(f_i)}$ along distinct edges $f_i$, then $Q(G[k]; w, \alpha, \beta, \gamma)=Q(G; w', \alpha, \beta, \gamma)$, where $w'$ agrees with $w$ away from the summed edges and, at each $f_i$, replaces the five per-edge parameters by the unique solution of the $5\times5$ linear system (15), whose right-hand side records the $Q$-values of the five one-edge reductions of $H^{(f_i)}$. The proof resolves each copy of $H^{(f_i)}$ by the five-term recursion defining $Q$; Lemma 2.6 identifies each resulting one-edge packaged arrow presentation with one of the five edge operations on $G$, and Lemma 2.3 lets those operations be carried out in any order. Corollary 3.7 restates the result as a tensor product formula for $Q$ in the shape of Brylawski's original formula. Setting $c=x=y=0$ gives the polynomial $Z$, and through Equations (5) and (6) this specialises to a Brylawski-style tensor product formula for the Krushkal polynomial; the Bollobás-Riordan case is Theorem 4.1, and the transition, ribbon graph, and Tutte polynomial formulas are further specialisations.

Load-bearing premise

The proof rests on two asserted-but-unproved structural facts: the five edge operations commute on distinct edges, and each operation equals a gluing ('2-sum') with one of five one-edge packaged arrow presentations; if either fails, the recursion that defines $Q$ and the tensor product formula would not hold.

Editorial extensions

If this is right

The Krushkal polynomial of a tensor product $G\otimes_\varphi H$ can be computed from $Q(G)$ after solving one $5\times5$ linear system whose data come from $H$ alone, as stated in Corollary 3.3 together with Equations (5) and (6).
The Bollobás-Riordan polynomial gains a tensor product formula, Theorem 4.1 and Corollary 4.2, that does not require $H$ to be plane or orientable; the older plane-case formula is recovered as a specialization in Corollary 4.3.
The same theorem specialises to the topological transition polynomial (Theorem 4.4), the ribbon graph polynomial, and the Tutte polynomial, so Brylawski's original formula is a special case.
Because the 2-sum is now defined through packaged arrow presentations, tensor products along loop edges, which were previously undefined or required compromises, are covered uniformly.
When properties of $H$ rule out some of the five reductions, the $5\times5$ system collapses to a smaller one, and this is the mechanism behind the simpler formulas found in earlier work.

Reading between the lines

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

An implicit consequence is computational: since the $5\times5$ substitution depends only on the second factor, a fixed $H$ gives a precomputed linear transformation, so tensor products with many copies of the same $H$ can be evaluated by one substitution per edge.
The five-operation recursion for $Q$ may be a master deletion-contraction template for other embedded-graph invariants; if so, packaged arrow presentations would be the natural normal form for proving Brylawski-type formulas for future polynomials defined by such recursions.
Because Penrose-contraction is the operation behind twisted duality for embedded graphs, the tensor product formula likely transfers to signed or dual variants of the Bollobás-Riordan polynomial, a direction the paper does not pursue.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Here's my take. The paper delivers the first Brylawski-style tensor product formula for the Krushkal polynomial and packages it with a framework that also recovers—and in some cases strengthens—the known formulas for the Bollobás–Riordan, ribbon graph, transition, and Tutte polynomials. The packaged arrow presentations and the auxiliary five-operation polynomial Q do real work: they handle loops and non-cellular embeddings, which is exactly where the earlier partial results from Huggett–Moffatt and Ellis-Monaghan–Moffatt got stuck. The proof of the main theorem, an induction on the number of 2-sums via one-edge reductions, is sound in outline, and the recovery of Brylawski's original formula as a specialization is a convincing sanity check.

The soft spots are real but not fatal. The paper leans on four lemmas (2.3–2.6) whose proofs are omitted or dismissed as 'straightforward but lengthy.' These are load-bearing: Lemma 2.3 is what makes Q well-defined and what justifies the coefficient matching in Theorem 3.6; Lemma 2.6 is what turns each 2-sum into one of the five operations. They are probably true—a finite exhaustive check on the one-edge packaged presentations would settle both—but the text does not let a reader verify them by reading alone. I'd want that check visible before calling the proof complete.

Smaller issues: the reference to 'Remark 3.1.2' matches no numbered remark (there is an unnumbered remark there), and the matrix in Corollary 4.2 looks garbled. Both are typo-level. The stress-test note's concern about the 5x5 matrix degenerating at alpha=1 or beta=1 is also real, but the reduced systems in Section 4 handle the specializations used in applications; the uniqueness claim is over the rational function field, so the generic nonsingularity is the right notion. The citation pattern is normal: the earlier results from the same group are used as tools (relations between Z and the Krushkal polynomial), not as inputs to the main induction.

Bottom line: a strong paper with a new, checkable result and a useful framework. The gaps are in exposition, not in the core idea. It deserves a serious referee. I'd encourage the editor to ask for a short verification of the deferred lemmas rather than desk-reject. I'd bring it to reading group and would cite it if I worked in this area.

Referee Report

4 major / 4 minor

Summary. The paper defines a tensor product for packaged arrow presentations (coloured ribbon graphs, equivalently graphs in pseudo-surfaces) and introduces a seven-variable polynomial Q satisfying a recursion with five edge operations: deletion, contraction, Penrose-contraction, merge-deletion, and merge-contraction. The main theorem (Theorem 3.6) gives a Brylawski-type tensor product formula for Q, from which formulas for the Krushkal polynomial and the Bollobás–Riordan polynomial, as well as known formulas for the topological transition polynomial, the ribbon graph polynomial, and the Tutte polynomial, are derived as corollaries. The paper positions this as the first Brylawski-style tensor product formula for the Krushkal polynomial and as a strict strengthening of earlier partial results for the Bollobás–Riordan polynomial.

Significance. If the gaps identified below were filled, the result would be a significant contribution: it unifies and generalizes several previously scattered tensor product formulas for topological Tutte polynomials, using a coherent packaged-arrow-presentation formalism. The paper explicitly recovers Brylawski's original formula, Huggett–Moffatt's formulas, and transition polynomial formulas, and the matrix-based statement of the tensor product formulas is concrete and usable. However, the central theorem is currently conditional on unproved operational lemmas and on an unexamined singular-locus issue in the linear systems, so the contribution is not yet fully established.

major comments (4)

[§2.3–§2.4, Lemmas 2.3, 2.4, 2.5, 2.6] The paper's main theorem rests on the commutativity of the five packaged operations (Lemma 2.3), the passage of these operations through packaged 2-sums (Lemma 2.5), and the realization of each operation by a 2-sum with a one-edge packaged presentation (Lemma 2.6). All three are asserted with proofs suppressed ("straightforward but lengthy", "we omit their proofs", and only the K5 case is illustrated). Since Lemma 2.3 is used explicitly in the proof of Proposition 3.4 to establish that Q is well-defined, and Lemma 2.6 is used to convert Equation (16) into Equation (17) in the proof of Theorem 3.6, these are load-bearing facts, not routine details. Complete proofs, or a detailed verification such as an appendix with exhaustive case analysis, are required.
[§3.1.2, Theorem 3.6, Eq. (15)] The assertion that the 5×5 matrix in Eq. (15) is non-singular and that the ϕ's are uniquely determined is false at several specializations. For α=1 the last three rows of the matrix coincide; in fact the determinant equals α^3(α−1)^2(β−1)(γ−1), so it also vanishes at β=1, γ=1, and α=0. Section 4 specializes to γ=1 in Eq. (20), and the Tutte polynomial corollary in §4.3 sets α=1, so the uniqueness claim does not hold at parameter values used later in the paper. The theorem needs either an explicit "for generic parameters" statement with a description of the singular locus, or a proof that the reduced systems used in Section 4 remain nonsingular on the relevant domains.
[§3.1.2, proof of Theorem 3.6, after Eq. (16)] The proof assumes that after applying the recursion (12) to all edges of H(fk) other than e(k), and removing created or isolated vertices, the remaining one-edge packaged arrow presentation is one of the K_i in Figure 8. However, Figure 8 is explicitly stated to show only cases with [u]V ≠ [v]V and [a]B ≠ [b]B. If e is a loop (u=v) or if operations on other edges merge the classes of the endpoints of e, the resulting one-edge presentation is not among the depicted K_i. Lemma 2.6 only asserts the direction "2-sum with K_i realizes the operation", not that every one-edge presentation is equivalent to some K_i. The proof therefore leaves unhandled the case of loop edges and of coincident vertex or boundary classes.
[§3.1.2, Eq. (15) and following computation] The derivation of the matrix entries in Eq. (15) is not shown. The text displays only the computation of Q(K1 \ e) = α^2β^2γ and then states that the five equations can be rewritten as the matrix equation. The analogous values for Q(K_i \ e), Q(K_i / e), Q(K_i ⋌ e), Q(K_i ∠ e), and Q(K_i ∠ e) for i = 2,...,5 are not given, so the reader cannot verify the matrix entries without reconstructing Figure 8 and the definition of Q from scratch. Please include these calculations explicitly.

minor comments (4)

[§4.1, Corollary 4.2] The matrix displayed in Corollary 4.2 appears to contain a stray γ in the second row (the entry "1αγ1 1"); it should presumably match the matrix of Theorem 4.1, where the second row is "1 α 1 1".
[§3.1.2, before Theorem 3.6] The phrase "satisfactorially completes" should be "satisfactorily completes".
[§3.1.2, "T echnique 1"] The formatting "T echnique 1" contains an extra space; more importantly, the technique is described in prose rather than as a formal lemma, which makes the reductions used in Section 4 (setting some ϕ's to zero) harder to verify.
[§4, throughout] The paper would benefit from a short discussion of the singular loci of the linear systems in (15), (22), and (30), since these systems are used to define substitutions that feed into the tensor product formulas and the corollaries specialize the parameters in ways that can hit the singular points.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 3.6 is a structural induction from the defining recursion of Q; the Brylawski-style formulas are specializations, not inputs. Self-citations support auxiliary identifications only, and the omitted commutativity proofs are a rigor gap, not a circular argument.

full rationale

The central tensor product formula (Theorem 3.6, Eqs. (14)-(15)) is derived by induction on k: the factor Q(H^{(f_k)}) is expanded against the five one-edge packaged arrow presentations K_1,...,K_5, Lemma 2.6 rewrites G[k-1] ⊕ K_i as the five edge operations on f_k, and Eq. (12) is then used in reverse to recognize Q(G[k-1]; w''). The coefficients φ are fixed by the 5x5 linear system whose right-hand sides are Q(H^{(f_i)}\e), Q(H^{(f_i)}/e), etc.; this is exactly the Brylawski mechanism (solve for coefficients from the H-factor, substitute them into G), not a quantity fitted from the target G ⊗ H. The later formulas for the Krushkal, Bollobás-Riordan, transition, and Tutte polynomials are obtained by specialization (c=x=y=0 for Z, w=(b_e,a_e,c_e,0,0) and α=t, β=γ=1 for the transition polynomial, etc.) or by recovery of previously known results, so no renaming of inputs as predictions occurs. The self-citations [15], [18] establish well-definedness of Z and the bridge from Z/T^{ps} to the Krushkal polynomial; these are external theorems, and the well-definedness of Q is proved internally in Proposition 3.4 (modulo Lemma 2.3), so the main claim does not reduce to a self-citation. The genuine caveat is rigor, not circularity: Lemma 2.3 ('a straightforward but lengthy calculation ... we omit the details'), Lemma 2.5 ('we omit their proofs'), and Lemma 2.6 ('The remaining cases are verified similarly and we omit the details') are load-bearing for Proposition 3.4 and Theorem 3.6; if commutativity or the K_i realizations failed, the recursion defining Q and the tensor product formula would collapse. That is a completeness risk to be checked, not a circular dependency.

Assumptions & free parameters 0 free parameters · 5 assumptions · 1 invented entities

All claims rest on the packaged arrow presentation formalism and on several lemmas whose proofs are omitted. No numerical parameters are fitted; the polynomial variables are formal indeterminates.

assumptions (5)

ad hoc to paper The five edge operations on packaged arrow presentations commute (Lemma 2.3).
Stated without proof; used to prove well-definedness of Q in Proposition 3.4 and in the main tensor product theorem.
ad hoc to paper 2-sums of packaged arrow presentations commute with edge operations and are associative (Lemmas 2.4 and 2.5).
Proofs omitted as straightforward but lengthy; needed for the tensor product definition and for Lemma 2.6.
ad hoc to paper Each of the five operations is realized by a 2-sum with a one-edge packaged arrow presentation K_i (Lemma 2.6).
Proof omitted; verified only for K5 in Figure 9; directly used in the induction step of Theorem 3.6.
domain assumption The polynomial Z is well-defined and relates to the Krushkal polynomial via Equations (4)-(6), from Huggett-Moffatt [15, Theorem 24, Theorem 29, Corollary 42].
Invoked in Section 3.1.1 to connect the tensor product formula for Q to the Krushkal and Bollobás-Riordan polynomials.
domain assumption Packaged arrow presentations correspond to coloured ribbon graphs and to graphs embedded in pseudo-surfaces, from Huggett-Moffatt [15].
Used in the Remark after Definition 4 to justify the geometric interpretation of the algebraic framework.

invented entities (1)

Polynomial Q(G;a,b,c,x,y,alpha,beta,gamma)
purpose: A five-term deletion-contraction polynomial whose tensor product formula specializes to Z, Krushkal, Bollobás-Riordan, transition, ribbon graph, and Tutte polynomial formulas.
Q is defined by a recursion in Proposition 3.1 and is needed because Z alone lacks relations for Penrose-contraction, merge-deletion, and merge-contraction. It is a formal mathematical device, not an empirically evidenced entity.

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Cite this review

Pith. "Pith review of Tensor product formulas for the Bollob\'as-Riordan and Krushkal polynomials." pith.science (2026). https://pith.science/paper/OQKRHF5F

@misc{pith2026250522570,
  author       = {Pith},
  title        = {Pith review of: Tensor product formulas for the Bollob\'as-Riordan and Krushkal polynomials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OQKRHF5F}},
  note         = {Machine review of arXiv:2505.22570}
}

read the original abstract

Brylawski's tensor product formula expresses the Tutte polynomial of the tensor product of two graphs in terms of Tutte polynomials arising from the tensor factors. Analogous tensor product formulas are known for the ribbon graph polynomial and transition polynomials of graphs embedded in surfaces, as well as for the Bollob\'as-Riordan polynomial in some special cases. We define the tensor product of graphs embedded in pseudo-surfaces and use this to generalize and unify all of the above results, providing Brylawski-style formulas for both the Bollob\'as-Riordan and Krushkal polynomials.

Figures

Figures reproduced from arXiv: 2505.22570 by the authors.

**Figure 2.** The 2-sum of G and H with respect to the coupling φ. shown in Figure 1a. Note that the operations of deletion, contraction and Penrose-contraction all commute with each other and themselves when applied to distinct edges. (This follows immediately from the observation that the operations act locally at edges.) They also correspond to the usual ribbon graph operations of the same name. 2.2 2-sums and tensor products … view at source ↗

**Figure 3.** Two arrow presentations, G and H, with a coupling φ, and the 2-sum G ⊕φ H. Then G ⊕φf,e H = H ⊕φ −1 f,e G and G ⊕φf,e (H ⊕φg,h K) = (G ⊕φf,e H) ⊕φg,h K. Lemma 2.2. Let G and H be arrow presentations, f be an edge of G, and e and g be distinct edges of H. Additionally let φ be a coupling of f and e. Then 1. (G ⊕φ H) \ g = G ⊕φ (H \ g), 2. (G ⊕φ H) / g = G ⊕φ (H / g), and 3. (G ⊕φ H) ⋌ g = G ⊕φ (H ⋌ g). We now make us… view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: Contracting and merge-deleting the edge f. • G Penrose-contract e, denoted G ⋌ e, is the packaged arrow presentation (G ⋌ e, V ′′′ , B ′′) where V ′′′ = (V − {[u], [v]}) ∪ {[u] ∪ [v] ∪ T − {u, v}}, where T is the set of vertices of G ⋌ e created by the Penrose-contract…

**Figure 5.** Figure 5: Naming conventions used in Definition 5. [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

**Figure 6.** Figure 6: Forming the 2-sum of packaged arrow presentations. [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]

**Figure 7.** Figure 7: Two packaged arrow presentations, G and H, with a coupling φ, and the 2-sum G ⊕φ H. • In G let u denote the vertex containing −−→p1p2; v denote the vertex in containing −−→p3p4; a denote the boundary component containing p1 and p4; and b denote the boundary component c…

**Figure 8.** Figure 8: The five packaged arrow presentations with one edge and no isolated vertices. Here [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]

**Figure 9.** Figure 9: Showing G ⊕φ K5 = G ∠ f. here φ is any coupling of f and e, where e is the singleton edge in the appropriate Ki . Proof. The result can be verified by comparing the resulting packaged arrow presentations on the left and right hand sides of each equation [PITH_FULL_IMA…

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Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages

[1]

M. Aigner. The Penrose polynomial of a plane graph.Math. Ann., 307(2):173–189, 1997

work page 1997
[2]

Askanazi, S

R. Askanazi, S. Chmutov, C. Estill, J. Michel, and P. Stollenwerk. Polynomial invariants of graphs on surfaces.Quantum Topol., 4(1):77–90, 2013

work page 2013
[3]

Bollob´ as and O

B. Bollob´ as and O. Riordan. A polynomial invariant of graphs on orientable surfaces.Proc. Lond. Math. Soc. (3), 83(3):513–531, 2001

work page 2001
[4]

Bollob´ as and O

B. Bollob´ as and O. Riordan. A polynomial of graphs on surfaces.Math. Ann., 323(1):81–96, 2002

work page 2002
[5]

Brylawski

T. Brylawski. The Tutte polynomial. I. General theory. InMatroid theory and its applications, pages 125–275. Liguori, Naples, 1982

work page 1982
[6]

C. Butler. A quasi-tree expansion of the Krushkal polynomial.Adv. Appl. Math., 94:3–22, 2018

work page 2018
[7]

S. Chmutov. Generalized duality for graphs on surfaces and the signed Bollob´ as-Riordan polynomial.J. Combin. Theory Ser. B, 99(3):617–638, 2009

work page 2009
[8]

J. A. Ellis-Monaghan and I. Moffatt. Twisted duality for embedded graphs.Trans. Am. Math. Soc., 364(3):1529–1569, 2012

work page 2012

Show all 23 references

[9]

J. A. Ellis-Monaghan and I. Moffatt.Graphs on surfaces. Dualities, polynomials, and knots. SpringerBriefs Math. New York, NY: Springer, 2013

2013
[10]

J. A. Ellis-Monaghan and I. Moffatt. A Penrose polynomial for embedded graphs.European J. Combin., 34(2):424–445, 2013

2013
[11]

J. A. Ellis-Monaghan and I. Moffatt. Evaluations of topological Tutte polynomials.Comb. Probab. Comput., 24(3):556–583, 2015

2015
[12]

J. A. Ellis-Monaghan and I. Sarmiento. Generalized transition polynomials.Congr. Numer- antium, 155:57–69, 2002

2002
[13]

J. A. Ellis-Monaghan and I. Sarmiento. A recipe theorem for the topological Tutte polynomial of Bollob´ as and Riordan.Eur. J. Comb., 32(6):782–794, 2011

2011
[14]

Huggett and I

S. Huggett and I. Moffatt. Expansions for the Bollob´ as-Riordan polynomial of separable ribbon graphs.Ann. Comb., 15(4):675–706, 2011

2011
[15]

Huggett and I

S. Huggett and I. Moffatt. Types of embedded graphs and their Tutte polynomials.Math. Proc. Cambridge Philos. Soc., 169(2):255–297, 2020

2020
[16]

F. Jaeger. On transition polynomials of 4-regular graphs. InCycles and rays (Montreal, PQ, 1987), volume 301 ofNATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., pages 123–150. Kluwer Acad. Publ., Dordrecht, 1990

1987
[17]

Jaeger, D

F. Jaeger, D. L. Vertigan, and D. J. A. Welsh. On the computational complexity of the Jones and Tutte polynomials.Math. Proc. Cambridge Philos. Soc., 108(1):35–53, 1990. 26

1990
[18]

Krajewski, I

T. Krajewski, I. Moffatt, and A. Tanasa. Hopf algebras and Tutte polynomials.Adv. in Appl. Math., 95:271–330, 2018

2018
[19]

Krushkal

V. Krushkal. Graphs, links, and duality on surfaces.Comb. Probab. Comput., 20(2):267–287, 2011

2011
[20]

C. Merino. Computational techniques. In J. Ellis-Monagan and I. Moffatt, editors,Handbook of the Tutte polynomial and related topics, pages 141–160. CRC Press, 2022

2022
[21]

I. Moffatt. Knot invariants and the Bollob´ as-Riordan polynomial of embedded graphs.Eur. J. Comb., 29(1):95–107, 2008

2008
[22]

Moffatt, S

I. Moffatt, S. Noble, and M. Thompson. Tensor products of multimatroids and a Brylawski- type formula for the transition polynomial. Preprint, arXiv:2309.00493, 2023

2023 arXiv
[23]

Thompson.Topological Analogues of the Tutte polynomial and their Decompositions

M. Thompson.Topological Analogues of the Tutte polynomial and their Decompositions. PhD thesis, Royal Holloway, University of London, 2024. 27

2024

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

M. Aigner. The Penrose polynomial of a plane graph.Math. Ann., 307(2):173–189, 1997

work page 1997

[2] [2]

Askanazi, S

R. Askanazi, S. Chmutov, C. Estill, J. Michel, and P. Stollenwerk. Polynomial invariants of graphs on surfaces.Quantum Topol., 4(1):77–90, 2013

work page 2013

[3] [3]

Bollob´ as and O

B. Bollob´ as and O. Riordan. A polynomial invariant of graphs on orientable surfaces.Proc. Lond. Math. Soc. (3), 83(3):513–531, 2001

work page 2001

[4] [4]

Bollob´ as and O

B. Bollob´ as and O. Riordan. A polynomial of graphs on surfaces.Math. Ann., 323(1):81–96, 2002

work page 2002

[5] [5]

Brylawski

T. Brylawski. The Tutte polynomial. I. General theory. InMatroid theory and its applications, pages 125–275. Liguori, Naples, 1982

work page 1982

[6] [6]

C. Butler. A quasi-tree expansion of the Krushkal polynomial.Adv. Appl. Math., 94:3–22, 2018

work page 2018

[7] [7]

S. Chmutov. Generalized duality for graphs on surfaces and the signed Bollob´ as-Riordan polynomial.J. Combin. Theory Ser. B, 99(3):617–638, 2009

work page 2009

[8] [8]

J. A. Ellis-Monaghan and I. Moffatt. Twisted duality for embedded graphs.Trans. Am. Math. Soc., 364(3):1529–1569, 2012

work page 2012