Polytopality of 2-orbit maniplexes

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Introduction
Abstract polytopes are posets with some properties that generalize those of the face-poset of convex and skeletal polytopes.Since their introduction by Schulte in [4], the main approach to study abstract polytopes has been by their symmetries.It is known that the group of symmetries (or automorphism group) of an abstract polytope acts freely on its flags: chains consisting of one element of each rank (dimension).Therefore, we may measure how symmetric a polytope is by counting how many flag orbits it has under the action of its automorphism group.A k-orbit polytope is one that has k flag orbits under this action.The smaller the number of flag orbits, the more symmetric the polytope.
We call 1-orbit polytopes regular, and they have been by far the most studied.
The book [10] is the standard reference and it is dedicated solely to abstract regular polytopes.
It is quite clear that 2-orbit polytopes are the second most symmetric kind of polytopes after the regular ones, so, naturally, they have been the second most studied family.Nevertheless, the general theory of 2-orbit polytopes has been much more challenging than that of regular ones.
One can classify 2-orbit polytopes of rank n in 2 n − 1 different symmetry types.These symmetry types are determined by a proper subset I ⊂ {0, 1, . . ., n − 1}, that in some way consists of the "permitted kinds of reflections", and are denoted by 2 n I .One of those symmetry types consists of the so called chiral polytopes: those with all possible "rotational" symmetry but no "reflection" symmetry at all (2 n ∅ ).Historically, chiral polytopes have been the most studied type of 2-orbit polytopes.The main theory of (abstract) chiral polytopes was developed in 1991 by Schulte and Weiss [18] but the existence of chiral polytopes in any rank was one of the main questions to consider.Rank 3 chiral polytopes had been studied in the context of maps on surfaces, and in the 1970's Coxeter gave examples of rank 4 chiral polytopes arising as quotients of hyperbolic tessellations [2].However, it is not until 2008 that algebraic methods were developed to find (finite) chiral polytopes of rank 5 [1].In 2010, almost 20 years after the publication of [18], Pellicer proved that there exist abstract chiral polytopes in any rank at greater than 2 [14].Also in 2010, Hubard [7] described a way to construct 2-orbit polyhedra of any symmetry type from groups; examples of 2-orbit polyhedra for each of the 7 possible symmetry types are well-known.In 2016, Pellicer [15] found geometric examples of 2-orbit polytopes of any rank with a fixed symmetry type.However, the challenge of finding 2-orbit polytopes of any possible type has not had that much of an advancement.
Maniplexes are a generalization of polytopes where some conditions on connectivity are relaxed.In 2019, Pellicer, Potočnik and Toledo [16] found a way to construct 2-orbit maniplexes of any type.To do this, they used a voltage assignment on the symmetry type graph.However, the question of whether or not their examples are flag-graphs of polytopes remained open.This is due to the fact that there was no method to find the intersection properties that the so called voltage group should satisfy for the derived maniplex to be polytopal.However, in [8], Hubard and the author of this paper described a method to find the intersection properties for every symmetry type for any number of orbits (see Theorem 3.2).Using this method for 2-orbit maniplexes, we have a way to test and determine their polytopality.
The 2-orbit n-maniplexes constructed in [16] depend on some choices: a regular (n − 1)-maniplex and a monodromy (to be defined in 2) of that maniplex, both satisfying some conditions.In their paper, Pellicer, Potočnik and Toledo proved that given a 2-vertex symmetry type, there exists a maniplex M and a monodromy η of M satisfying such conditions and thus, the n-maniplex obtained has the desired symmetry type graph.In this paper we shall show that some of the maniplexes constructed in [16] are in fact polytopal.More specifically, we shall see that if {0, 1, . . ., n − 1} \ I has exactly two elements, there are choices for the maniplex M and the monodromy η that ensure that the maniplex constructed in [16] with symmetry type 2 n I is polytopal.The first sections of this paper will introduce all the concepts that we will use.
In Section 2 we define the basic concepts such as abstract polytopes, flags, maniplexes, premaniplexes, automorphisms, symmetry type graphs, monodromies and coverings.In Section 3 we define the concepts of fundamental groupoid and fundamental group of a graph, voltage graphs, derived graphs, and we enunciate Theorem 3.2, which will be our main tool.In Section 4 we describe the construction 2M , which gives rise to a family of regular polytopes of all ranks that we will be using for all our examples.In Section 5 we describe the construction given in [16] to get 2-orbit maniplexes, and we also add some observations that will be helpful when proving the main result.Finally, in Section 6 we use this construction and Theorem 3.2 to prove that if I has exactly n − 2 elements, then there exists a polytope P with symmetry type 2 n I .

Basic concepts
An abstract polytope is defined as a poset (P, ≤) that is flagged, strongly flagconnected and satisfies the diamond property.We clarify each of these proper- The unique flag i-adjacent to Φ is denoted Φ i .
We say that a flagged poset is strongly flag-connected if given two flags Φ, Ψ, there is a sequence Φ 0 = Φ, Φ 1 , Φ 2 , . . ., Φ k = Ψ, such that Φ j−1 and Φ j are i j -adjacent for some i j satisfying that Φ and Ψ have different i j -faces.
From now on, we will refer to abstract polytopes just as polytopes.We will also abuse notation and call P an abstract polytope assuming that the symbol ≤ will be used for the order relation (and < for the strict order relation).A polytope of rank n is called an n-polytope.
Given a polytope P, the flag-graph of P, denoted G(P), is the edge-colored graph whose vertices are the flags of P and i-adjacent flags are joined by an edge of color i.The flag-graph of an n-polytope is an example of what is called an n-maniplex or a maniplex of rank n.That is, a connected simple graph with edges colored with the numbers {0, 1, . . ., n − 1}, in such a way that: • every vertex is incident to an edge of each color, and • given two colors i and j such that |i − j| > 1, the alternating paths of length 4 using edges of these two colors are closed.
The vertices of a maniplex are usually referred to as flags.
At this point, we point out that although infinite maniplexes and polytopes can be very interesting, for the purposes of this paper, all maniplexes and polytopes are assumed to be finite.Some of the results and definitions may also apply for the infinite case, but not all of them (particularly, the main definition of Section 4 needs some fine-tuning for the infinite case).
Given a set of colors I ⊂ {0, 1, . . ., n − 1}, and an n-maniplex M, we use the notation M I for the subgraph of M induced by the edges with colors in I.
We will denote the complement of I in {0, 1, . . ., n − 1} by I, and if I = {i}, we denote its complement just by i.The connected components of M i are the i-faces of M.
We say that an i-face F and a j-face G of a maniplex are incident if they have non-empty intersection.If in addition i ≤ j, we write F ≤ G.
In this way we have defined a poset P(M).If P is a polytope, then P(G(P)) is isomorphic to P. On the other hand, in [6] it is proven that if M satisfies certain conditions called path intersection properties, then P(M) is a polytope and G(P(M)) is isomorphic to M, showing that such maniplexes, called polytopal, are exactly the flag-graphs of polytopes.
An isomorphism of n-maniplexes is just a graph isomorphism that preserves the colors of the edges.Naturally, an automorphism of a maniplex is just an isomorphism onto itself.The automorphism group of a maniplex M is denoted by Γ(M).It is known that the automorphism group of a maniplex acts freely on its flags (the proof is identical to that of [10, Proposition 2A4], which is the particular case of abstract polytopes).Moreover, the automorphisms of the flag-graph of a polytope correspond to the automorphisms of the polytope itself (as a poset) (see [10,Lemma 2A3] for one inclusion.The other inclusion can be proved using the construction in [6, Section 3]).
The symmetry type graph (STG) of a maniplex M, denoted by T (M), is simply the quotient of M by its automorphism group.That is, the vertices of T (M) are the flag orbits of M under its automorphism group, and two orbits are connected by an edge of color i if there is a pair of i-adjacent flags, one in each of those orbits.If two i-adjacent flags are on the same orbit, we draw a semi-edge of color i on the vertex corresponding to that orbit.Notice that T (M) is not necessarily simple, but it satisfies the other conditions that define an n-maniplex: it is connected, each vertex is incident to exactly one edge of each color in {0, 1, . . ., n − 1}, and the alternating paths of length 4 using two non-consecutive colors are closed.We call such a graph an n-premaniplex, or just a premaniplex if the rank n is implicit.
The symmetry type graph of a polytope P is just the symmetry type graph of its flag-graph, and it is denoted by T (P).
A k-orbit polytope (or maniplex) is one with exactly k flag orbits under the action of its automorphism group.In other words, one whose symmetry type graph has exactly k vertices.A 1-orbit polytope (or maniplex) is called regular.
Given a maniplex M, we define r i as the flag permutation that maps each flag to its i-adjacent flag.It is easy to see that r i is an involution and that, if |i − j| > 1, then r i r j is also an involution.In particular r i and r j commute whenever i and j are not consecutive.
The group Mon(M) = ⟨r i : 0 ≤ i ≤ n − 1⟩ is called the monodromy group of M, and we will call each of its elements a monodromy.It is worth mentioning that some authors prefer the term connection group and the notation Con(M).
An alternative way of defining the automorphism group of a maniplex M is as the set of flag permutations that commute with the elements of the monodromy group.
For this paper, both the automorphism and the monodromy group will act on the right.
If M is a regular maniplex (sometimes called reflexible), for a fixed flag Φ we can define ρ i as the (unique) automorphism that maps Φ to Φ i .It is known that {ρ i : i ∈ {0, 1, . . ., n − 1}} is a generating set for the automorphism group of M (see [10,Theorem 2B8] for the polytopal case).Moreover, the function mapping ρ i to r i can be extended to a group anti-isomorphism between Γ(M) and Mon(M) (see, for example, [12,Theorem 3.9] or [20,Section 7]).In particular, the monodromy group of a regular maniplex acts regularly on its flags.
Given two n-premaniplexes M and X , a covering projection (covering for short) from M to X , is a function p from the vertices of M to the vertices of X that preserves i-adjacencies.It can be proved that all coverings are surjective.
If e is an edge M connecting vertices x and y, we define p(e) as the edge connecting p(x) and p(y) that has the same color as e.If there is a covering from M to X , we say that M covers X .Note that if M is a maniplex and X is its symmetry type graph, the natural projection from M to X is always a covering.

Voltage graphs and intersection properties
A dart in a graph is just a directed edge.The inverse of a dart d, denoted by d −1 is the dart that corresponds to the same edge with the opposite orientation.We think of semi-edges as having only one orientation, having thus only one dart which is inverse to itself.An edge with two different endpoints (and therefore, with two different darts) will be called a link.Every dart d has an initial vertex, or start-point I(d), and a terminal vertex or endpoint T (d), having the property that its underlying edge is incident to both I(d) and T (d), and that For more on graphs with semi-edges see [9], for example.
A path is a sequence of darts We say that a path is closed if its start-point and endpoint are equal.Otherwise, it is open.
We also consider a formal empty path for each vertex that goes from that vertex to itself.If W is the empty path from v to v, we abuse notation and denote it by v as well.
Two paths W and W ′ with the same start-point and endpoint are said to be homotopic if one can transform W into W ′ by a finite sequence of the following operations: • Inserting two consecutive inverse darts at any point, that is where • Deleting two consecutive inverse darts at any point, that is In this case we write W ∼ W ′ .
It is known that homotopy is an equivalence relation and that it is preserved by the usual concatenation of paths.We often abuse notation and use W as the name for a path or for its homotopy class.Furthermore, even if we refer to W as a path, we will always be thinking of it up to homotopy.
The set of all homotopy classes of paths in a graph X, together with the concatenation operation forms a groupoid called the fundamental groupoid of X, which we will denote by Π(X).The subset of closed paths based at a vertex x forms a group denoted by Π x (X) and called the fundamental group of X based at x.It is easy to show that if X is connected, all its fundamental groups are isomorphic (moreover, they are conjugates by elements of the fundamental groupoid).
Given a spanning tree T and a vertex of a graph X we can find a distinguished set of generators for the fundamental group Π x (X).For each dart d in X but not in T , take the path C d that goes from x to the starting point of d through T , then takes d, and then goes back to x from the endpoint of d trough T .It is easy to see that Π x (X) is generated by {C d }, where d runs among the darts in X not in T .
Given a graph X and a group Γ, a voltage assignment (with voltage group Γ) To construct a voltage assignment, it is enough to define it for the darts of X (in such a way that inverse darts have inverse voltages).Then, the voltage where d runs among the darts of the graph X not in a given spanning tree T .This will be important later, since very often we want to know the voltages of some sets of closed paths.Knowing the voltages of a fundamental group will also help us calculate the set of voltages of open paths with fixed start-point and endpoint as a coset of the voltages of a fundamental group.
Given a voltage graph (X, ξ) with voltage group Γ, we can construct the derived graph X ξ as follows: • The vertex set is V × Γ where V is the vertex set of X.
• The dart set is D × Γ where D is the dart set of X.
• If the dart d goes from x to y, the dart (d, γ) goes from the vertex (x, γ) to (y, ξ(d)γ).
• (Optional) If a dart (or vertex) d has a color c, then the dart (or vertex) Note that the inverse of the dart (d, γ) is the dart The voltage group Γ always acts by automorphisms on the derived graph X ξ , with the action given by (x, γ)σ = (x, γσ), where x is a dart or vertex of X and γ, σ ∈ Γ.It should also be clear that if X has some coloring of its vertices or darts, then Γ preserves the induced coloring when acting on X ξ .
In [8], Hubard and the author of this paper prove the following results: Let X be a premaniplex and ξ : Π(X ) → Γ a voltage assignment such that X has a spanning tree with trivial voltage on all its darts.Then X ξ is a maniplex if and only if the following conditions hold: • The set ξ(D) generates Γ where D is the set of darts of X .
• If d is the dart of a semi-edge, then ξ(d) has order exactly two. • In this paper we will use Theorem 3.2 to show that we can construct 2-orbit polytopes with some prescribed symmetry type graphs.
To be more precise, the conditions on M are (equivalent to) the following: 1. M has to cover X n (the premaniplex of rank n obtained by deleting the edges of color n from X ) 1 .
2. There is an involutory monodromy η in M that maps all the flags of any given facet to different facets.
A concrete family of maniplexes satisfying these conditions is given and its elements are called M n where n denotes the rank.This family is constructed recursively and it does not depend on the choice of X .More concretely, M 2 is the flag-graph of the square, and M n+1 is constructed from M n as 2Mn (which we will define shortly).In [13] it is proved that if P is a regular polytope then 2G(P) is the flag-graph of a regular polytope, and we will see shortly a proof of the fact that if M is regular 2M is regular too.This implies that the family The construction 2M works as follows: Given a maniplex M, the flags of , where F(M) is the set of flags of M and Fac(M) is the set of facets.We will think of Z as the set of functions from Fac(M) to the cyclic group Z 2 .Then the adjacencies in 2M are defined by: where Fac(Φ) denotes the facet of Φ and given a facet F ∈ Fac(M) the vector χ F is the one associated with the characteristic function of F , that is, the vector with 1 in the coordinate corresponding to F and 0 in every other one.
1 In [16] this is called having a coloring consistent with I where X = 2 n+1 I .
If we remove the edges of color n from 2M we get one connected component . Each component consists of all flags of type (Φ, x) where Φ is a flag of M. In particular, every facet of 2M is isomorphic to M.
Given a polytope P we can construct a polytope 2P such that the flag-graph of 2P is the maniplex 2G(P) .This construction is the following: the support of f , denoted by supp(f ), is defined as the set of facets F ∈ Fac(M) If we denote by Fi the set of i-faces of 2P , then for / ∼ where (F, x) ∼ (F ′ , x ′ ) if and only if F = F ′ and for every facet G ∈ supp(x + x ′ ) we have that F ≤ G.We then add a formal greatest face F n+1 .Finally, the incidence relation on 2P is given by (A, x) < (B, y) if and only if A < B and (A, x) ∼ (A, y).
We should remark that the construction 2P is actually (2 P * ) * , where the superscript * denotes duality and 2 P is a better-known construction (see [3]).
In [5] it is mentioned that the operation 2M where M is a maniplex generalizes 2P where P is a polytope.What the authors mean by this is that if P is a polytope, then G( 2P ) is isomorphic to 2G(P) (see [11,Section 5.1] for an elementary proof).
The automorphisms of M have a natural action on 2M .First we define a left action of Γ(M) on Z Fac(M) 2 as follows: given an automorphism σ of M, a vector x in Z and a facet F of M we define that σx : f → x(f σ).Now, if σ is an automorphism of M, we can extended it to an automorphism of 2M by defining (Φ, x)σ := (Φσ, σ −1 x).Note that supp(σ −1 x) = supp(x)σ, so we are actually using the natural action of an automorphism on a set of facets.
In [5,Section 6] it is proved that Γ(M) this action on 2M is by automorphisms.Moreover, the automorphism group of 2M is a product of Γ(M)and } .This implies that if an n-maniplex M has symmetry type graph X then the symmetry type graph of 2M is obtained by adding semi-edges of color n to each vertex of X .In particular, if M is regular, then 2M is regular as well.
When we let M 2 be the flag-graph of a square and define M i+1 := 2M i for i ≥ 2, M 3 happens to be the flag graph of the map on the torus called {4, 4} (4,0) which can be thought of as a 4 × 4 chess board in which we identify opposite sides (without twisting).The 3-faces of every subsequent M n will be of this type and we will make use of this in our proofs.
A lattice is a poset in which every pair of elements and there is at least one facet of 2P containing both (A, x) and (B, y) then, where C is the greatest lower bound of the set supp(x + y) ∪ {A, B} .For a full proof of a more general result see [17,Theorem 5].
Corollary 4.1.For n ≥ 2, the poset associated with M n is a lattice.
In [16,Proposition 15,Lemma 17] it is proved that the family {M n } n≥3 satisfies both conditions mentioned at the beginning of this section.To prove that it satisfies the second condition, the authors prove and use the following lemma: Lemma 4.2.[16, Lemma 16] If M has a set of facets S which is not invariant under any non-trivial automorphism, then there is an involutory monodromy η of 2M that maps all the flags of any given facet to different facets.) is a set of facets of 2M and it satisfies the same condition.
In [16] the authors only care that S is not invariant under any non-trivial automorphism, however we shall choose an S that satisfies some extra conditions, which will prove useful when dealing with the polytopality of the constructed 2-orbit maniplexes.
From now on, we call the shaded set in Figure 1 S 3 , and we call S n ⊂ Fac(M n ) the set constructed recursively as S n := Ŝn−1 for n ≥ 4. Note that S 3 is not invariant under non-trivial automorphisms of M 3 , and therefore S n is not invariant under non-trivial automorphisms of M n .In [16], the authors actually use the complement of the set we have chosen as S 3 .The reason why we have chosen the complement will be apparent when we prove Lemma 4.4.
Definition 4.3.Given a polytope P and a face f ∈ P let us define the closure of f , denoted by f , as the set of all the facets of P which are incident to f .Given a pre-ordered2 set (P, ≤), one can give it a topology by defining that a subset C is closed if and only if whenever x ∈ C and x ≤ y we get that y ∈ C. In fact every topology on a finite set can be obtained from a pre-order in this way (see, for example, [19]).In such context, the closure of a set S is Proof We proceed by induction on n.For n = 3 this is a simple observation obtained from Figure 1.Note that the closure of a face is contained in the closure of any incident face of smaller rank, so we only need to prove the lemma for 0-faces.Suppose the lemma is true for M n .Let u, v be 0-faces of M n+1 , so that u = (u ′ , x) and v = (v ′ , y) for some 0-faces u ′ , v ′ of M n and x, y ∈ Z Fac(Mn) 2 .
We will prove that u ∪ v does not contain even S n+1 \ {(F n , 0)}, so we may assume that both u and v are each incident to at least one element of S n+1 other than (F n , 0), say for example that u = (u ′ , x) < (F n , χ G ) for some facet G of M n .By our definition of the order <, this implies that u ′ < F n (which is tautological) and (u ′ , x) ∼ (u ′ , χ G ), so we may assume without loss of generality that x = χ G .Analogously we may assume that there exists a facet In topological terms, Lemma 4.4 tells us that no set of two elements is dense in S n .
Now we turn our attention to the monodromy η.In the proof of Lemma 4.2 found in [16], η is constructed as follows: Let M be a regular maniplex and let S be a set of facets not invariant under non-trivial automorphisms.For every facet F ∈ S let Φ F be a fixed base flag in that facet.Let F 0 be a base facet in S and let Φ = Φ F 0 .For every F ∈ S, let ω F be a monodromy of M that maps Φ to Φ F .Note that since M is regular, its monodromy group acts regularly on its flags, so ω F is actually unique.Then for every flag (Ψ, x) in 2M define (Ψ, x)η := (Ψ, x + F ∈S χ Fac(Ψω F ) ).The action of η will be more clear with the following lemma.
Lemma 4.5.Let (Ψ, x) ∈ 2M .Let γ be the automorphism of M mapping the base flag Φ of F 0 to Ψ.Then, the vector corresponding to the facet of (Ψ, x)η differs from x only in the coordinates corresponding to Sγ, that is, if Proof For every F ∈ S we have that By doing the change of variable G = F γ we get y = x + G∈Sγ χ G and supp(x + y) = Sγ.
Given two sets S and S ′ of facets of a polytope P, we say that S ′ is a copy of S if there is an automorphism γ of P such that S ′ = Sγ.Lemma 4.5 tells us that if (Ψ, x)η = (Ψ, y) then supp(x + y) is a copy of S.
In [16,Lemma 16] it is proved that η is in fact a monodromy of 2M and that if S is not invariant under non-trivial automorphisms of M then η maps all the flags of a facet of 2M to different facets.
Note that using a set of facets of M n not invariant under non-trivial automorphisms we constructed the monodromy η for M n+1 .We have found such sets of facets for M n with n ≥ 3, which means that we have found the monodromy η for n ≥ 4. We have not found the monodromy η for M 3 , since every set of facets (edges) of M 2 (the square) is invariant under some non-trivial automorphism.Actually M 3 does not have such a monodromy.
However, the map on the torus {4, 4} (8,0) (a chess board of regular size where each border is identified with its opposite) does admit such a monodromy: simply take η = r 2 r 1 r 0 r 1 r 2 r 1 r 2 r 1 .This monodromy is indeed involutory (use the fact that r 1 r 2 has order 4) and it maps the 8 flags of a facet to the 8 squares where a knight could legally move in a game of chess.This example also covers all 2-vertex premaniplexes in rank 3, so, to construct 2-orbit polytopes of rank 4, one may use this example instead of M 3 .

2-orbit maniplexes
The symmetry type of a 2-orbit maniplex is denoted by 2 n I where n denotes the rank and I is a proper subset of {0, 1, . . ., n − 1}.Some authors assume the rank is implicit and just write 2 I , but we shall not do that in this paper as we will be working with several different ranks at once.In this notation, I is the set of colors i such that any flag is in the same orbit as its i-adjacent flag.
In terms of the symmetry type graph, I is the set of colors of the semi-edges.
Informally, we may think of I as the set of "allowed kinds of reflections".In particular, chiral n-polytopes are those of type 2 n ∅ , that is, those in which no reflections are allowed.
In order to find 2-orbit polytopes (resp.maniplexes) with every possible 2vertex premaniplex as its symmetry type graph, we only need to find examples with all the symmetry types 2 n I where 0, n − 1 ∈ I, and then use the operations 2M and duality (see [16,Corollary 12] for details).So without loss of generality, we will assume that 0, n − 1 ∈ I.
Given a 2-vertex premaniplex X = 2 n+1 I with 0, n ∈ I, the authors of [16] give a voltage assignment to its darts such that the derived graph is a 2-orbit maniplex with symmetry type graph isomorphic to X .We discuss this voltage assignment next.
Let us color the vertices of X one white and one black.Let {r 0 , r 1 , . . ., r n−1 } be the distinguished generators of the monodromy group of M n .If d is a semiedge incident on the white vertex we assign to it the voltage r i where i is its color.If d is a semi-edge of color i incident to the black vertex, we assign to it the voltage r 0 r i r 0 , or in other words, r i if i > 1 and r 0 r 1 r 0 if i = 1.Finally, if d is a dart of color i < n from the white vertex to the black vertex we assign to it the voltage r 0 r i , in particular, the voltage of the edge of color 0 is trivial.We shall give the voltage of the edge of color n shortly (which will be an involution, so orientation is irrelevant).From now on, we call this voltage assignment ξ.
Let p : M n → X n be a covering (remember that X n is the result of removing the edge of color n from X ).We color the flags of M n white or black according to the color of their image under p.Since X n has exactly 2 vertices and M n covers X , then every monodromy of M n either preserves the color of every flag or it changes the color of every flag.Note that the voltages of every dart preserve the color of the flags when thought of as monodromies.

The voltage of the darts of color n
Now we turn our attention to the voltage of the darts of color n.In [16] the authors make the voltage group act on (M n ) w × Z 2ℓ for some large ℓ, where (M n ) w is the set of white flags of M n (recall that all voltages preserve the color of the flags).This action is simply defined by (Ψ, x)ω = (Ψω, x) for all (Ψ, x) ∈ (M n ) w × Z 2ℓ and every monodromy ω.In other words, each monodromy acts as usual in the (M n ) w -coordinate and as the identity on the Z 2ℓ -coordinate.
The voltage y n of the darts of color n is defined as the composition of three commuting involutions ρ 0 , r 0 and s.Both r 0 and ρ 0 act only on the (M n ) w coordinate and are independent of the Z 2ℓ one, while s acts only on the Z 2ℓ coordinate.To avoid future confusion, the reader must recall that The importance of the Z 2ℓ coordinate lies in that it is used to prove that the derived maniplex is in fact a 2-orbit maniplex (and not a regular one), but it has no relevance when proving that the derived graph is a maniplex.For our purposes, we will ignore s for now.Therefore, we may think of all actions as being on , and assign the voltage y n := ρ 0 r 0 to the edge of color n.
In order to define ρ 0 we first need to choose a base flag Φ F for each facet F of M n .For the proof in [16] to work, we first choose a base facet of M n (where f n−1 is the greatest face of M n−1 ) and then for every white flag Ψ in F 0 we set Ψr 0 ηr 0 to be the base flag of its facet.Recall that η maps all the flags of F 0 to different facets, so there is no ambiguity.In [16] all the other base flags are chosen arbitrarily, however we will later have a preferred choice for them too.Then we define ρ 0 as the flag-permutation that acts on each facet F as the reflection (facet automorphism) that fixes all the faces of the base flag Φ F but its 0-face.
Note that if we replace a base flag with any flag sharing the same edge (1face) and the same facet, we get the same permutation ρ 0 .So we may define a base edge of a facet F as the 1-face of the corresponding base flag and forget about the base flag.
Now the previous choice of base edges would be equivalent to the following: For every black flag Ψ in F 0 we set (Ψη) 1 (the edge of Ψη) to be the base edge of its facet.
Let Ψ be a black flag in F 0 .Since M n = 2M n−1 , and by the choice of F 0 , the flag Ψ can be written as (ψ, 0) for some flag ψ in M n−1 .Lemma 4.5 tells us that Ψη = (ψ, x) for some x ∈ Z So for facets corresponding to vectors whose support is a copy of S n−1 we are forced to choose a specific base edge, but for any other facet we may choose the base edge as we want.Let (e, 0) be the base edge of the base facet Then for every x whose support is not a copy of S n−1 we choose (e, x) as the base edge of the facet (f n−1 , x).We summarize this in the following definition: Definition 5.1.Base edges: Let (e, 0) be the base edge of the base facet of and let f n−1 be the greatest face of M n−1 .If supp(x) is not a copy of S n−1 , then we define the base edge of the facet (f n−1 , x) of M n to be (e, x).If supp(x) = S n−1 γ for some γ ∈ Γ(M n−1 ), then we define the base edge of (f n−1 , x) to be (eγ, x).
is such that supp(x) ⊂ u ∪ v for some faces u, v in M n−1 , then the base edge of the facet (f n−1 , x) is (e, x).
Corollary 5.3.If (e, 0) is the base edge of the base facet it is also the base edge of any other facet containing it.
Finally we let y n := ρ 0 r 0 and extend the voltage assignment ξ by assigning y n as the voltage of the edge of color n of X .Note that since ρ 0 acts as an automorphism in each facet, it commutes with all the monodromies that do not use the generator r n .In particular ρ 0 commutes with r 0 , implying that y n is an involution.
As previously discussed, in [16] the authors consider the voltage group as acting on (M n ) w × Z 2ℓ where (M n ) w is the set of white flags of M n and ℓ is some large integer.If (Ψ, a) ∈ (M n ) w × Z 2ℓ and ω is a color preserving monodromy of M n then (Ψ, a)ω is simply defined as (Ψω, a).Then they use y n := ρ 0 r 0 s as the voltage of the edge of color n, where s is an involution acting only on the Z 2ℓ coordinate.We will call the voltage assignment used in [16] ξ ′ .In [16] the authors prove that X ξ ′ is a maniplex with symmetry type graph isomorphic to X .The proof of the fact that X ξ ′ is a maniplex also applies to X ξ , but the proof of the fact X ξ ′ is not regular relies heavily on ℓ being large.
On the other hand X ξ is either regular or has symmetry type graph isomorphic to X .However our proofs to show polytopality of some derived maniplexes will be much clearer if we work with the voltage assignment ξ and consider that If d m is a dart from the black vertex to the white one, then, since but since c m ̸ = 1 we know that r cm r 0 = r 0 r cm and we get And if d m is a semi-edge on the black vertex then We want to show that if X is the 2-vertex premaniplex with links of colors 0 and n and semi-edges of the colors in-between (see Figure 3), and ξ is the voltage assignment defined in (2), then the condition of Theorem 3.2 is satisfied, that is, we want to prove that if for all k, m ∈ [0, n] and all pairs of vertices (a, b).
In [16] the authors only use the sets S n to find the monodromy η of M n mapping all the flags of a given facet to different facets.To ensure that η acts r i r 0 i r r 0 y n n i i 0 this way they only need to use the fact that S n is not invariant under non-trivial automorphisms, and they do not consider any other properties.However, for our purposes this condition is not enough.We also need η to map the flags of the base facet "very far away".This is to ensure that "close facets" have the same base edge as the base facet.This is why we have proved Corollary 5.2.
First we will prove that the intersection condition is satisfied for k > 1 for every 2-vertex premaniplex 2 n+1 I where 0, n ∈ I.
Theorem 6.4.Let X be an (n+1)-premaniplex with two vertices and with links of color 0 and n, that is X = 2 n+1 I with 0, n ∈ I. Let ξ be the voltage assignment defined in (2).Then, for k > 1 and for all m ∈ [0, n] we have that for all pairs of vertices (a, b) in X .
Before proceeding with the proof, let us introduce a few new concepts.Let M be a maniplex of rank n and let µ be a flag-permutation of M. Let i ∈ {0, 1, . . ., n − 1} and let F and G be i-faces of M. If µ maps every flag with i-face F to a flag with i-face G we will say that µ maps F to G.In the case where F = G we will say that µ fixes F .If µ fixes F for every i-face F we will say that µ fixes i-faces.M n−1 , which is naturally isomorphic to any facet of M n (see Figure 4).
If F = (f n−1 , x) is a facet of M n which has (e, x) as its base edge, then y n interchanges flags with 1-face (e 0 , x) with flags with 1-face (e 1 , x), while it fixes flags with 1-face (e, x) (see Figure 5).
), let F be a facet with base edge (e, x), and let Ψ be a flag with facet F and 1-face (e j , x) for j ∈ {0, 1}.Let (ψ, y) = Ψω.If we write ω as a product of the voltages of darts, every time we change the facet of M n (that is, every occurrence of r n−1 or r 0 r n−1 ), we must change to a new facet with the same edge.This means that the edge of Ψω must be the same as the edge of Ψ, or in other words, that (e j , y) ∼ (e j , x) where ∼ is the equivalence relation we used when defining 2P for a polytope P. Then supp(x + y) is contained in  facet to one where the support of the corresponding vector differs in coordinates corresponding to a set contained in e 0 ∪ e 1 .Let We claim that supp(x i ) ⊂ e 0 ∪ e 1 .This is proved by a simple induction on i.
For i = 1 we have that x 1 = 0, which has support supp(0) = ∅ ⊂ e 0 ∪ e 1 .If the claim is true for i, since supp(x i ) ⊂ e 0 ∪ e 1 , Corollary 5.2 tells us that the facet of M n corresponding to x i has (e, x i ) as its base edge.Hence, if (ψ i , x i ) has (e j , x i ) as its 1-face, then (ψ i+1 , x i+1 ) = (ψ i , x i )y n ω i+1 has e 1−j as its 1-face.This implies that supp(x i + x i+1 ) ⊂ e 0 ∪ e 1 .Then Thus we have proved our claim.Note that x = x s , so our claim and Corollary 5.2 tell us that the facet (f n−1 , x) of M n (where f n−1 is the greatest face of M n−1 ) has base edge (e, x).
) 0 with its voltage assignment.
Figure 6).Then, the generator corresponding to a semi-edge of color on the white vertex is its voltage r i .The generator corresponding to a semi-edge e on the black vertex is the voltage of the path a n ea −1 n , where a n is the dart from the white vertex to the black one with color n.This voltage is But since ρ 0 acts as an automorphism in each facet, for i < n − 1 we get ρ 0 r i ρ 0 = r i , and we get ξ(Π a  Now, if m = n − 1 we have a little more work to do.We proceed as in the previous case, but we cannot ensure that x = 0.However, we have that Φω = (ψ, x) and the inclusion supp(x) ⊂ u∪v still holds.By writing the 2-face of Φ as (Φ) 2 = (Q, 0), we know that Q is a square.Let w be the opposite vertex of u in Q and q be the opposite vertex to v, so that the vertices of Q are uvwq in cyclical order (see Figure 7).
Here we have used the way Γ(M n−1 ) acts on M n discussed on Section 4. Note that we have used the same symbol ρ 1 to denote an automorphism of M n and also an automorphism of M n−1 , but because of the way Γ(M n−1 ) acts on M n this is actually not ambiguous.
On the other hand, the vector corresponding to Φ 1 is 0, so Lemma 6.7 tells us that supp(ρ 1 x) ⊂ u ∪ v. Every facet incident to both v and q must be incident to Q = v ∨ q (see Corollary 4.1), and hence also to u.Then, the intersection orbit symmetry types exist, in particular those with 1 or 2 links.Applying the constructions 2 P or 2P repeatedly to the examples of 2 links of Theorem 6.8 or the examples of rank 3 one gets that all symmetry types with 1 or 2 links exist in any rank higher than two.In addition to this, as previously discussed, in [14] Pellicer proves that chiral polytopes (those with symmetry type 2 n ∅ ) exist in rank 3 and higher, and if we apply the constructions 2 P or 2P to those, we get all symmetry types where all the links have consecutive colors.In conclusion, we have the following theorem: Theorem 6.9.Let n ≥ 3 and let X = 2 n I be a 2-vertex premaniplex of rank n.Let I := {0, 1, . . ., n − 1} \ I be the set of the colors of the links of X .Then, in any of the following cases, X is the symmetry type graph of a polytope.
• I has exactly 1 or 2 elements.
• I is an interval [k, m] = {k, k + 1, . . ., m}.Theorem 6.9 ensures that of the total of 2 n − 1 premaniplexes of rank n with 2 vertices, at least n 2 −n+1 are the symmetry type of a polytope (n with 1 link, n(n−1) 2 with 2 links and n(n−1)

2
− n + 1 with an interval of links of size at least 3).
It appears that there is still a long way to go.Nevertheless, Theorem 6.4 and Corollary 6.5 ensure that to prove that there are polytopes with symmetry type 2 n I , one should only check that (4) is satisfied for k = 1 for the voltage ξ (and this would imply that it is also satisfied for ξ ′ ).Sadly, the proof of Theorem 6.8 cannot be easily generalized to arbitrary I.This is because the voltage of a link of color 1 ≤ i < n would be r 0 r i and this affects most arguments used, mainly because there would be paths whose voltage change 0-faces but they do not use the color 0. The author of this paper conjectures that if this challenge is solved for one example 2 n I with 3 or more links of non-consecutive colors, the same solution must work for all others, and that would prove that there exist 2-orbit polytopes of any possible symmetry type (and rank n ≥ 3).
flagged if it has a least element F −1 , a greatest element F n , and all maximal chains, called flags, have the same finite size.If the size of the flags is n + 2, we say that the flagged poset has rank n.Moreover, given an element F of a flagged poset, if the maximal chains having F as their biggest element have r + 2 elements, we say that F has rank r and write rank(F ) = r.For the purposes of this paper, an element of rank i will be called an i-face, and in general, the elements of a flagged poset will be called faces.The (n − 1)-faces in a flagged poset of rank n are called facets.If Φ is a flag, we denote its face of rank i by (Φ) i .We say that a flagged poset satisfies the diamond condition if given two faces F, G such that F < G and rank(G)−rank(F ) = 2, there exist exactly two faces H 1 , H 2 such that F < H i < G for i ∈ {1, 2}.We say that two flags Φ and Ψ in a flagged poset are i-adjacent if they have the same faces with ranks different than i but they have different i-faces.If a flagged poset satisfies the diamond condition, every flag Φ has exactly one i-adjacent flag for i ∈ {0, 1, . . ., n − 1}.
If d and d ′ share both their start-point and their endpoint, then ξ(d) ̸ = ξ(d ′ ).• If |i − j| > 1 every (closed) path W of length 4 whose darts alternate between these two colors has trivial voltage.Before giving the next result we need to introduce some notation.Given two integers k and m we define [k, m] := {k, k + 1, . . ., m}.If k > m, then we define [k, m] as the empty set.If X is an n-premaniplex, a, b are two vertices in X , and I ⊂ [0, n − 1], then we define Π a,b I (X ) as the subset of Π(X ) consisting of all the paths from a to b that only use darts with colors in I.In particular, Π a,b [k,m] (X ) is the set of paths from a to b that only use colors between k and m.Theorem 3.2.[8, Theorem 4.2] Let X be a premaniplex of rank n and ξ : Π(X ) → Γ a voltage assignment.Then X ξ is polytopal if and only if it is a maniplex and the following equation holds for all k, m ∈ {0, 1, . . ., n − 1} and all pairs of vertices a, b ∈ X : of flag-graphs of regular polytopes.Since there are known examples of 2-orbit polyhedra (rank 3) with any given symmetry type, we are only concerned about the family {M n } n≥3 , which are the maniplexes used to construct 2-orbit maniplexes of ranks 4 and higher.

Figure 1 :
Figure 1: The set S 3 consisting of the shaded 2-faces is not invariant under any non-trivial symmetry of M 3 and it is not contained in the closure of two 0-faces.
just S = {y ∈ P : ∃x ∈ S, x ≤ y}.If we remove the least and greatest faces of a polytope P and equip it with this topology, then what we are calling the closure of f is actually just the set of facets contained in the topological closure of {f }.Lemma 4.4.Given any two proper faces u, v of M n with n ≥ 3, the set S n is not contained in u ∪ v.

.Lemma 6 . 2 .Lemma 6 . 3 .
If we repeatthis argument for d m−1 , then d m−2 and so on, and note that since c i ̸ = 1 then r c i commutes with r 0 for all i, we get the desired result.If Φ is a flag and K ⊂ {0, 1, . . ., n − 1}, we will denote by (Φ) K the set of faces in Φ whose rank is in K.The following lemma characterizes the voltages of closed paths that do not use edges of color 1 or the edge of color n and voltage y n .Let ω be a monodromy of M n that preserves the color of its flags and K ⊂ {0, 1, . . ., n − 1}.Suppose 1 ∈ K. Then for every white flag Φ of M n we have that (Φ) K = (Φω) K if and only if ω is the voltage of a closed path based on the white vertex of X that does not use colors in K ∪ {n}.Proof If (Φ) K = (Φω) K , by strong connectedness of M n there is a path W from Φ to Φω not using colors in K. Let c 1 c 2 • • • c k be the sequence of colors of W . Then Φω = Φr c 1 r c 2 • • • r c k , but since M n isregular, the action of the monodromy group on the flags is regular, so we conclude that ω= r c 1 r c 2 • • • r c k .Let W be the path on X that starts on the white vertex and follows thesequence of colors c 1 c 2 • • • c k , that is W = p( W ).We know that Φω and Φ are both white, so W must be a closed path.Recall that the voltage of W is the product of the voltages of its darts but in reverse order.Let a 1 a 2 • • • a k be the dart sequence of W .Let us consider W −1 = a k a k−1 • • • a 1 .Since W does not use colors in K and 1 ∈ K, Claim 6.1 tells us that ξ(W −1 ) = r ε 0 ω and since W is closed, W −1 is closed too, so ε = 0 and ξ(W −1 ) = ω.Thus we have found a closed path that does not use colors in K ∪ {n} and has voltage ω.For the converse, let W = a 1 a 2 • • • a k be a closed path based on the white vertex of X and suppose that W does not use colors in K ∪ {n}.Since W does not use the color 1 (because 1 ∈ K), Claim 6.1 tells us that ω := ξ(W ) =r c k r c k−1 • • • r c 1 where c i is the color of a i .Since c i / ∈ K ∪ {n} we know that (Φω) K = (Φ) K .Using the same logic one can prove the following lemma, which characterizes the voltages of open paths that do not use edges of color 1 or the edge of color n and voltage y n .Let ω be a monodromy of M n that preserves the color of the flags and K ⊂ {0, 1, . . ., n − 1}.Then (Φ) K = (Φ 0 ω) K if and only if ω is the voltage of an open path in X that does not use colors in K ∪ {n}.

ΦFigure 4 :
Figure 4: The edges e, e 0 and e 1 illustrated on a 3-face of M n−1 .

Figure 5 :
Figure 5: If (e, x) is the base edge of a facet, then y n interchanges the edges (e 0 , x) and (e 1 , x) while it fixes the edge (e, x).

Figure 7 :
Figure 7: The 2-face Q and its vertices u, v, w and q.