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Regular genus of $\mathbb{S}^2 \times \mathbb{S}^1 \times \mathbb{S}^1$, $4$-torus, and small covers over $\Delta^2 \times \Delta^2$

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Pith's one-line read This paper proves exact regular genera for three families of 4-manifolds: $G(\mathbb{S}^2 \times \mathbb{S}^1 \times \mathbb{S}^1)=6$, $G(\mathbb{S}^1 \times \mathbb{S}^1 \times \mathbb{S}^1 \times \mathbb{S}^1)=16$, and $G=8$ for each of…

desk verdict Solid new results on regular genus with a load-bearing unproved move; worth refereeing after the Observation in §2.2 is proved. read the letter →

arxiv 2506.01315 v1 pith:AUXQBHHN submitted 2025-06-02 math.GT math.CO

Anshu Agarwal , Biplab Basak This is my paper
classification math.GTmath.CO MSC 57Q1557S2552B1152B7005C15
keywords $\mathbb{Z}_2^n$-actionsmallcoverD-JequivalencesimplepolytopecrystallizationregulargenusPL4-manifoldtorus
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

Every closed PL 4-manifold can be encoded as an edge-colored graph called a crystallization, and the regular genus of the manifold is the smallest surface genus into which such a graph embeds regularly. This paper pins down that invariant for three families: it proves the regular genus of $\mathbb{S}^2 \times \mathbb{S}^1 \times \mathbb{S}^1$ is 6, confirming a conjecture left open in the literature, and proves the regular genus of the 4-torus is exactly 16, closing a gap between previously known bounds. It then classifies the small covers over $\Delta^2 \times \Delta^2$ up to D-J equivalence, showing there are exactly seven, and proves each has regular genus 8, including the product $\mathbb{RP}^2 \times \mathbb{RP}^2$. The values are obtained through explicit vertex-minimal crystallizations with 40, 120, and 52 vertices, respectively, whose regular genera are computed from bi-colored cycle counts and matched against a known lower bound. The paper also proposes that the regular genus of the $n$-torus is $1+\frac{(n+1)!(n-3)}{8}$ for $n \ge 5$.

What carries the argument

The central objects are crystallizations: 5-regular edge-colored graphs on the color set $\{0,1,2,3,4\}$ dual to contracted triangulations of a 4-manifold. The regular genus of a crystallization is computed from the bi-colored cycle counts $g_{ij}$ through the formula $\rho_\varepsilon(\Gamma) = 1 - \tfrac{1}{2}\big(-\tfrac{3}{2}|V(\Gamma)| + \sum_i g_{\varepsilon_i\varepsilon_{i+1}}\big)$ for a cyclic permutation $\varepsilon$ of the five colors; minimizing over all $\varepsilon$ and all crystallizations gives $G(M)$. The upper bounds come from explicit graphs, built from gems of $M\times \mathbb{S}^1$ using polyhedral glue moves and a two-stage dipole cancellation stated as an Observation in Section 2.2; the lower bound is the inequality $G(M) \ge 2\chi(M)+5m-4$, where $m = \mathrm{rk}(\pi_1(M))$.

What would settle it

Take the intermediate graph produced after the first reduction in the 4-torus construction (the claimed 156-vertex gem) and compute its fundamental group; if it is not the free abelian group of rank 4, the unproved Observation fails and the 120-vertex graph would not be certified. Similarly, checking the 80-vertex and 64-vertex intermediate gems for each small cover against the known $\mathbb{RP}^2$-bundle structure would settle whether the reduction to 52 vertices is valid.

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Extended reading notes

Core claim

The paper's central claim is that the regular genus, a PL invariant measuring the minimal genus of a surface in which a manifold's crystallization embeds regularly, takes the values $G(\mathbb{S}^2 \times \mathbb{S}^1 \times \mathbb{S}^1)=6$, $G(\mathbb{S}^1 \times \mathbb{S}^1 \times \mathbb{S}^1 \times \mathbb{S}^1)=16$, and $G(M)=8$ for each of the seven small covers over $\Delta^2 \times \Delta^2$. For each manifold it constructs a crystallization attaining the lower bound $2\chi + 5m - 4$ of Proposition 1, where $m$ is the rank of the fundamental group, and then computes the regular genus of that crystallization by counting bi-colored cycles with respect to a chosen cyclic permutation of the five colors. The 40-vertex and 120-vertex graphs settle the previously open cases of $\mathbb{S}^2 \times \mathbb{S}^1 \times \mathbb{S}^1$ and the 4-torus; the small-cover analysis classifies all $\mathbb{Z}_2$-characteristic functions on $\Delta^2 \times \Delta^2$ up to D-J equivalence, identifies one of the seven as $\mathbb{RP}^2 \times \mathbb{RP}^2$ and the other six as $\mathbb{RP}^2$-bundles over $\mathbb{RP}^2$, and supplies a 52-vertex crystallization for each. Along the way the paper records that the seven covers collapse to at most four distinct PL homeomorphism types, with pairwise identical isomorphism signatures, and that all crystallizations constructed are weak semi-simple, a property equivalent to attaining the lower bound.

Load-bearing premise

The argument rests on an unproved technical claim, stated as an Observation in Section 2.2, that certain graph-surgery moves used to shrink the crystallizations always preserve the underlying 4-manifold; if any of those moves goes wrong, the vertex reductions in the main theorems could produce graphs of the wrong manifold.

Editorial extensions

If this is right

  • The previously open conjecture that $\mathbb{S}^2 \times \mathbb{S}^1 \times \mathbb{S}^1$ has regular genus 6 is settled, so this manifold joins the known orientable prime PL 4-manifolds of genus 6 alongside the mapping tori constructed in the literature.
  • The regular genus of the 4-torus is exactly 16, closing the interval between the earlier bounds of 4 and 28 and providing an explicit 120-vertex crystallization.
  • Every small cover over $\Delta^2 \times \Delta^2$ has regular genus 8 and admits a 52-vertex genus-minimal crystallization; in particular $G(\mathbb{RP}^2 \times \mathbb{RP}^2)=8$.
  • The seven D-J equivalence classes of small covers over $\Delta^2 \times \Delta^2$ are at most four distinct PL homeomorphism types, with three pairs identified by identical isomorphism signatures.
  • All crystallizations constructed in the paper are weak semi-simple, so the class of closed PL 4-manifolds with known regular genus now includes these manifolds together with all of their connected sums.
  • The paper proposes that the regular genus of the $n$-torus is $1+\frac{(n+1)!(n-3)}{8}$ for $n \ge 5$, supported by the observed cycle structure of a candidate crystallization with $(n+1)!$ vertices.

Reading between the lines

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

  • If the unproved Observation in Section 2.2 holds generally, the two-stage 2-dipole/3-dipole reduction is a reusable tool: other products $M \times \mathbb{S}^1$ with a matching base crystallization could receive the same vertex reduction, potentially yielding genus-minimal crystallizations for further 4-manifolds.
  • The pairwise agreement of isomorphism signatures shows that D-J equivalence is strictly finer than PL homeomorphism in dimension four; a natural next question, not answered here, is whether the four remaining classes are pairwise non-homeomorphic.
  • The $n$-torus conjecture is already testable at $n=5$: the candidate graph has 720 vertices and would have regular genus 361, so the bottleneck is finding a matching lower bound for manifolds whose fundamental group has rank greater than two.
  • Because weak semi-simplicity is equivalent to attaining the regular-genus lower bound and is closed under connected sum, every new weak semi-simple example enlarges a class whose additivity over connected sums would resolve the 4-dimensional smooth Poincar\'e conjecture if it ever covered all simply-connected PL 4-manifolds.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper constructs explicit edge-colored graphs (crystallizations) for S^2×S^1×S^1 (40 vertices), for S^1×S^1×S^1×S^1 (120 vertices), and for each of the seven Davis-Januszkiewicz equivalence classes of small covers over Δ^2×Δ^2 (52 vertices each). For each constructed graph the authors compute the regular genus and combine it with the external lower bound G(M) ≥ 2χ(M)+5m−4 from Proposition 1 (Basak–Casali) to conclude G(S^2×S^1×S^1)=6, G(S^1×S^1×S^1×S^1)=16, and that each of the seven small covers, including RP^2×RP^2, has regular genus 8. The paper also proposes a conjectural formula for the regular genus of the n-torus, verifies the n=4 case by an independent cube-based construction, and reports Regina isomorphism signatures for the principal crystallizations.

Significance. If the constructions are correct, the paper settles a conjecture from Basak's 2019 mapping-torus paper and the previously open value for the 4-dimensional torus, and it gives the first regular-genus computations for small covers over Δ^2×Δ^2. The main strengths are the fully explicit graph constructions, the absence of free parameters in the upper-bound constructions, and the provision of machine-readable Regina isomorphism signatures that allow independent combinatorial verification of the final graphs. These are concrete, checkable contributions to crystallization theory and to the classification of PL 4-manifolds by regular genus.

major comments (4)
  1. [Subsection 2.2 (Observation)] The Observation is load-bearing and is not proved. After canceling the 2-dipole {v1, v1′}, the cancellation joins, for every color c not in {i, j}, the former c-neighbor of v1 to the former c-neighbor of v1′. This can create new paths between v2 and v2′ in the color set Δ4\{i,j,k}; the hypothesis that v1, v2, v1′, v2′ lie in distinct components of Γ_{Δ4\{i,j,k}} does not by itself imply the required distinct-component condition for v2 and v2′ in (Γ1)_{Δ4\{i,j,k}}. The assertion that v2 and v2′ form a 3-dipole after the first cancellation therefore needs a proof. The move is used repeatedly in Theorem 2 (moves Φ4, Φ5, Φ6) and Theorem 4 (moves Φ4 through Φ12); if the condition fails in any of these configurations, the reduced graphs may no longer be crystallizations of the claimed manifolds. Please supply a proof of the Observation or, failing that, a certified computational check of the dipole hypotheses after each intermediate step in Theorems 2 and 4.
  2. [Theorems 2 and 4] The Regina isomorphism signatures identify the final edge-colored graphs combinatorially, but an isomorphism signature alone does not certify which 4-manifold a graph represents; it certifies only that two graphs with the same signature are isomorphic. For Theorem 4, the later cube-based construction of Γ′ gives independent evidence that the 120-vertex graph represents the 4-torus, but for Theorem 2 (and for the identification of the small covers in Theorem 7) there is no independent recognition of the represented manifold. The manifold identification therefore rests entirely on the composition of moves in Sections 3 and 4, which is why the gap in the Observation is central. Please provide an independent recognition certificate for the 40-vertex graph of Theorem 2, for example by verifying that its associated colored triangulation is isomorphic to a known triangulation of S^2×S^1×S^1, or by constructing the final graph directly from a known gem.
  3. [Theorem 7] The proof of Theorem 7 is a long case analysis whose verification is delegated to Figure 9, with the sentence 'any statement whose justification is not explicitly provided can be understood to follow from Figure 9.' The final cycle counts—g{0,3}=g{0,4}=g{1,4}=g{2,3}=13 and g{1,2}=12 for the relevant permutations—are asserted after four polyhedral glue moves, but the reader cannot reconstruct all seven characteristic functions and all intermediate graphs from the figure without effectively repeating the whole computation. Since these numbers enter directly into the genus computation, please provide a systematic derivation or a machine-readable certificate (edge lists of all Γ′i, i=1,...,7, together with a script that computes the bi-colored cycle counts).
  4. [Theorem 7 and Lemma 6] The lower bound G(M(λ)) ≥ 8 uses Proposition 1 with m=2, the rank of the fundamental group. The statement 'it is easy to verify that the rank of the fundamental group of M^4(λ) is 2' is not backed by an argument or a reference. For the product RP^2×RP^2 the rank is 2, but for the six non-trivial RP^2-bundles the fundamental group should be computed explicitly or cited from a verifiable source. If any of these manifolds had rank 1, the lower bound would drop below 8 and the minimality conclusion would not follow. Please add the computation of π1 for each of the seven small covers.
minor comments (5)
  1. [Section 1] There are typos: 'dimesional' should be 'dimensional' and the journal name in reference [20] is 'Aequationes', not 'Acquationes'.
  2. [Theorem 2] In the sentence introducing the sets for the second reduction, the last set is written as 'Λ′4 = {vA′6, vA′7}' but it should be 'Λ′5'; as written, Λ′4 is defined twice.
  3. [Theorem 4] The cycle count for G′2 is stated as 'all the cycles colored by {i,j} are of length 4 and thus g{i,j}=30' for five color pairs, but the counting argument from Figure 8 is not shown. Adding the explicit count of 4-cycles for each pair would make the computation reproducible.
  4. [Theorem 7] The sentence about {2,3}-colored cycles in Γ′i is initially confusing: it says there are 13 four-cycles for i∈{1,2,5} and then 9 four-cycles, 2 six-cycles, and 2 two-cycles for i∈{3,4,6,7}. Both statements give total g=13, but the wording suggests a contradiction; please rephrase to make the totals explicit.
  5. [Figures 5–8 and 10–15] The figures are dense and use many labels; providing the edge lists of the final crystallizations in a supplementary file would make the Regina signatures and the cycle counts independently checkable without transcription from the figures.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the regular genus values are established by explicit crystallizations plus the external lower bound of Proposition 1; the unproved Observation in §2.2 is a proof gap, not a circular step.

full rationale

The central claims are not circular. Upper bounds are obtained by explicit edge-colored graph constructions: for S²×S¹×S¹ the paper starts from the standard 8-vertex crystallization of S²×S¹, builds a 64-vertex gem of S²×S¹×S¹, applies polyhedral glue moves and the §2.2 Observation to reach a 40-vertex crystallization, and then computes the regular genus directly from the bi-colored cycle counts g{i,j}. For the 4-torus it similarly starts from a 24-vertex crystallization of S¹×S¹×S¹, builds a 192-vertex gem, reduces it to 120 vertices, and computes ρε = 16 from the displayed cycle counts. For the small covers, the 96-vertex gems are constructed from the explicit colored triangulations in Figure 9 and then reduced to 52-vertex crystallizations; the genus 8 is computed from the listed cycle counts. In each case the lower bound comes from Proposition 1, an external published result (Basak–Casali, Forum Math. 2017), not from the paper's own constructions or fitted values. The equality between the upper bound and the lower bound is therefore a genuine match, not a prediction forced by construction. The paper does use the authors' prior work [1] for the small-cover gem construction, but the construction itself is reproduced in Figures 9–15, so the citation is not doing unverified load-bearing work. The main weakness is the unproved Observation in Subsection 2.2: it asserts that a 2-dipole removal followed by the induced 3-dipole removal yields a crystallization of the same 4-manifold, and this move is used repeatedly in Theorems 4 and 7. The preservation of the 3-dipole condition after the first cancellation is not proved, and the Regina signatures certify the final graphs combinatorially rather than the intermediate moves. This is a genuine correctness/verification gap, but it is not circularity: the Observation is not defined in terms of the target genera, nor does it fit any parameter to the desired answers, nor does it reduce the theorems to the assumptions of the theorems. Under the circularity criteria, no specific step qualifies as self-definitional, fitted-input-called-prediction, load-bearing self-citation, imported uniqueness, ansatz-smuggling, or renaming of a known result. The appropriate finding is therefore no significant circularity, with the proof-gap concern recorded as a correctness risk rather than as circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are introduced; the constructions are vertex-count specific. The central results depend on an external lower bound, a prior construction from the authors' own work [1], and an unproved observation about dipole moves.

assumptions (4)
  • standard math Proposition 1: for a closed connected PL 4-manifold M with rk(π1(M))=m, G(M) ≥ 2χ(M)+5m−4.
    Used as the lower bound to prove minimality in Theorems 2, 4, and 7. It is external to this paper.
  • ad hoc to paper The Observation in Subsection 2.2: under the stated hypotheses, removing a 2-dipole and then the induced 3-dipole yields a crystallization of the same manifold.
    Stated without proof and used to justify the vertex-reducing moves in Theorems 4 and 7.
  • standard math The construction of gems of small covers from [1] correctly yields colored triangulations dual to the small covers M⁴(λ_i).
    Used in Section 4 to build the 96-vertex gems; accepted from prior work by the same authors.
  • domain assumption For n=1,2,3,4 the graph Γ′ constructed in Section 3 represents the n-dimensional torus.
    Verified with Regina for n=4 and asserted for n≤3; the general n≥5 case is left as Conjecture 5.

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Pith. "Pith review of Regular genus of $\mathbb{S}^2 \times \mathbb{S}^1 \times \mathbb{S}^1$, $4$-torus, and small covers over $\Delta^2 \times \Delta^2$." pith.science (2026). https://pith.science/paper/AUXQBHHN

@misc{pith2026250601315,
  author       = {Pith},
  title        = {Pith review of: Regular genus of $\mathbbS^2 \times \mathbbS^1 \times \mathbbS^1$, $4$-torus, and small covers over $\Delta^2 \times \Delta^2$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AUXQBHHN}},
  note         = {Machine review of arXiv:2506.01315}
}
abstract

A crystallization of a PL manifold is an edge-colored graph encoding a contracted triangulation of the manifold. The concept of regular genus generalizes the notions of surface genus and Heegaard genus for 3-manifolds to higher-dimensional closed PL manifolds. The regular genus of a PL manifold is a PL invariant. Determining the regular genus of a closed PL $n$-manifold remains a fundamental challenge in combinatorial topology. In this article, we first resolve a conjecture by proving that the regular genus of $\mathbb{S}^2 \times \mathbb{S}^1 \times \mathbb{S}^1$ is 6. Additionally, we determine that the regular genus of $\mathbb{S}^1 \times \mathbb{S}^1 \times \mathbb{S}^1 \times \mathbb{S}^1$ is 16. We also present some observations related to the regular genus of the $n$-dimensional torus and conjecture that the regular genus of $\mathbb{S}^1 \times \mathbb{S}^1 \times \cdots \times \mathbb{S}^1$ ($n$ times) is $1+\frac{(n+1)! \ (n-3)}{8}$, for $n\ge 5$. Then, we investigate the regular genus of small covers. Small covers are closed $n$-manifolds admitting a locally standard $\mathbb{Z}_2^n$-action with orbit space homeomorphic to a simple convex polytope $P^n$. For the polytope $P = \Delta^2 \times \Delta^2$, we classify all the small covers up to Davis-Januszkiewicz (D-J) equivalence and show that there are exactly seven such covers. Among these, one is $\mathbb{RP}^2 \times \mathbb{RP}^2$, while the others are $\mathbb{RP}^2$-bundles over $\mathbb{RP}^2$. Remarkably, each of these seven small covers has the regular genus 8. Results in this article provide explicit regular genus values for several important 4-manifolds, offering new insights and tools for future work in combinatorial topology.

Figures

Figures reproduced from arXiv: 2506.01315 by the authors.

Figure 1
Figure 1. Standard 8-vertex crystallization of S 2 × S 1 . Let (Γ, γ) be a 2p-vertex crystallization of the 3-manifold M. Let us name the vertices of Γ by v0, v1, · · · , v2p−1. Since ∆3 is the color set, we denote (Γ, γ) by Γ(0, 1, 2, 3). For a, b, c ∈ ∆4, by Γ(a, b, c, ), we mean a 3-regular colored graph isomorphic to Γ(0, 1, 2, ), where the vertex map is the iden￾tity map and color map C : {0, 1, 2} → {a, b, c} is defined… view at source ↗
Figure 2
Figure 2. Construction of the gem G of M × S 1 using the crystallization of the 3-manifold M. Theorem 2. A crystallization of S 2 × S 1 × S 1 with 40 vertices realizes the minimum possible regular genus, which is 6. Proof. For constructing the crystallization of S 2 × S 1 × S 1 , let us take Γ(0, 1, 2, 3) to be the standard 8-vertex crystallization of S 2 × S 1 ( [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Crystallization moves on the gem G1. S 2 × S 1 × S 1 with 44 vertices, say G2 1 (Figure 3c). Again, Λ6 = {v D′ 0 , vD′ 1 },Λ ′ 6 = {v A′ 4 , vA′ 5 } satisfy conditions of the observation in Subsection 2.2. Thus, applying the move with respect to (Φ6,Λ6,Λ ′ 6 , {0, 2}) on G2 1 , we get the crystallization G′ 1 of S 2 × S 1 × S 1 with 40 vertices (Figure 3d). Note that the subgraph of G′ 1 generated by the remaining v… view at source ↗
Figures from the paper (12 more)
Figure 4
Figure 4. Figure 4: 24-vertex crystallization, Γ = (a) ∪ (b), of S 1 × S 1 × S 1 . Remark 3. The classification of PL n-manifolds by regular genus is a well-established problem in combinatorial topology. For orientable PL 4-manifolds, the classification is complete up to regular genus 5 (…
Figure 5
Figure 5. Figure 5: The subgraphs A′ , B′ , C′ , D′ of the gem G2. Proof. For constructing this crystallization, we take Γ(0, 1, 2, 3) to be the 24-vertex crystal￾lization of S 1 × S 1 × S 1 in [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: Subgraphs of the crystallization G1 2 . ure 8). The subgraph of G′ 2 generated by the remaining vertices of A′ , B′ , C′ , D′ is isomorphic to Γ(0, 1, 2, ). In Γ(0, 1, 2, 3), g{i,j} = 4 and all the cycles colored by {i, j} are of length 6, when {i, j} ∈ {{2, 3}, {1, 2}…
Figure 7
Figure 7. Figure 7: Subgraphs of the crystallization G2 2 . Consider the 2-cube. Let us represent it by (a 0 1 , a1 2 , a1 3 , a2 4 ). Here a 1 3 = a 0 1×I and a 2 4 = a 1 2×I, where I = [0, 1]. By a j i , we mean that it is the i th vertex of the cube and it is colored by the color j. Co…
Figure 8
Figure 8. Figure 8: Crystallization G′ 2 of S 1 × S 1 × S 1 × S 1 with 120 vertices. an (n + 1)-regular colored graph. Join two vertices vi and vj of Γ by a 0-colored edge if these are connected via a path n, n − 1, n − 2, · · · , 1, 2, 3, · · · , n of the colored edges in Γ. Thus, we get…
Figure 9
Figure 9. Figure 9: t j w with all its 3-faces and their Z2-characteristic vectors, for 1 ≤ j ≤ 6. 15 [PITH_FULL_IMAGE:figures/full_fig_p015_9.png]
Figure 10
Figure 10. Figure 10: The gem Γ1 of RP2 × RP2 with 96 vertices. Using the construction introduced in [1], we will construct gems of these small covers. Let us fix some notations that will be used throughout this section. For an element g = (c1, c2, c3, c4) in Z 4 2 , let the corresponding …
Figure 11
Figure 11. Figure 11: Subgraph S1 of Γ1. 6) [PITH_FULL_IMAGE:figures/full_fig_p017_11.png]
Figure 12
Figure 12. Figure 12: Compact form of Si , for 1 ≤ i ≤ 7. regular genus 8, for 1 ≤ i ≤ 7. In the proof of Theorem 7, any statement whose justification is not explicitly provided can be understood to follow from [PITH_FULL_IMAGE:figures/full_fig_p018_12.png]
Figure 13
Figure 13. Figure 13: The gem Γ1 1 of RP2 × RP2 with 80 vertices. that T 1 w0 ∈ Λ1. Let A2 be the {0, 4}-colored four-cycle containing T 6 0 and let Λ2 be the set of vertices of A2. Apply the second polyhedral glue move with respect to (Φ2,Λ2,Λ ′ 2 , 2) ( [PITH_FULL_IMAGE:figures/full_fig…
Figure 14
Figure 14. Figure 14: The gem Γ2 1 of RP2 × RP2 with 64 vertices. of the vertices of the form T j w, j ∈ {2, 4} and w is an entry from the fourth row of the compact form of S. Due to this polyhedral glue move on Γ1 , we get the gem Γ2 of M4 (λ) with 64 vertices, and all the {3, 4}-colored …
Figure 15
Figure 15. Figure 15: The crystallization Γ′ 1 of RP2 × RP2 with 52 vertices. colored by {0, 3}. In applying the third polyhedral glue move, 4 four-cycles colored by {0, 3} are removed. Thus, in Γ2 , there are 16 four-cycles colored by {0, 3}. When applying the fourth polyhedral glue move,…

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