A zonotope and a product of two simplices with disconnected flip graphs

We give an example of a three-dimensional zonotope whose set of tight zonotopal tilings is not connected by flips. Using this, we show that the set of triangulations of $\Delta^4 \times \Delta^n$ is not connected by flips for large $n$. Our proof makes use of a non-explicit probabilistic construction.


Introduction
We consider the poset P of polyhedral subdivisions of a polytope P or zonotopal tilings of a zonotope Z, ordered by refinement. This poset is called the Baues poset of P or Z. The minimal elements of this poset are, respectively, the triangulations of P or the tight zonotopal tilings of Z. Two minimal elements of P are connected by a flip if there is an element of P whose only proper refinements are these two minimal elements. The flip graph of P or Z is the graph whose vertices are minimal elements of P and whose edges are flips.
For zonotopal tilings, the flip graph is known to be connected for cyclic zonotopes [16] or if the zonotope has dimension two [6]. Our first result is the first example, to the author's knowledge, of a zonotope whose flip graph is not connected. This answers a question of Reiner in [9]. Our example is a three-dimensional permutohedron with many copies of each of its generating vectors. The number of copies of each vector is determined by a probabilistic argument to be around 100.
Using a related construction, we also show that the flip graph of triangulations of the product of two simplices is not generally connected. Santos [10] gave an example of a polytope whose flip graph of triangulations is not connected. However, the case when P is product of two simplices remained of special interest due to the appearances of these triangulations in various branches of mathematics; see [5,Chapter 6.2] for an overview. Santos [12] proved that the flip graph of ∆ 2 × ∆ n is connected for all n, and the author [7] proved that the flip graph of ∆ 3 × ∆ n is connected for all n. However, we show that the flip graph of ∆ 4 × ∆ n is not connected for n ≈ 4 · 10 4 . Connectivity of flip graphs is related to the generalized Baues problem, which concerns the topology of the poset P. Specifically, the problem asks if the order complex of P minus its maximal element is homotopy equivalent to a sphere. The question was formulated by Billera et al. in [2] and is motivated by the fact that when one restricts P to coherent subdivisons or zonotopal tilings, the resulting poset is isomorphic to the face lattice of a certain polytope, called the fiber polytope [1]. See [9] for an overview.
In general, the answers to the generalized Baues problem and the question of connectivity of flip graphs do not imply each other. However, there are situations where the answer to one can be used to answer the other. The generalized Baues problem for triangulations was answered in the negative by Santos [13] by using a point set in general position with disconnected flip graph. The problem for zonotopal tilings remains open.
The paper is organized as follows. Section 2 reviews triangulations and the product of two simplices. Section 3 reviews mixed subdivisions, zonotopal tilings, and the Cayley trick. Section 4 constructs our zonotope and proves that its flip graph is not connected. Section 5 proves that the flip graph of ∆ 4 × ∆ n is not connected. Section 6 is an appendix proving several propositions used in the paper.

Triangulations and the product of two simplices
We begin with a quick overview of triangulations, flips, and the product of two simplices. We refer to De Loera et al. [5] for a more comprehensive treatment.

Subdivisions and triangulations
Throughout this section, let A ⊂ R m be a finite set of points. A cell of A is a subset of A.
A simplex is a cell which is affinely independent. A face of a cell C is a subset F ⊆ C such that there exists a linear functional φ ∈ (R) m such that F is the set of all points which minimize φ on C. For any cell C, let conv(C) denote the convex hull of C. Definition 2.1. A polyhedral subdivision, or subdivision, of A is a collection S of cells of A such that 1. If C ∈ S and F is a face of C, then F ∈ S .
2. If C, C ′ ∈ S , then conv(C) ∩ conv(C ′ ) = conv(F ) where F is a face of C and C ′ .

C∈S conv(C) = conv(A).
The subdivision {A} is the trivial subdivision. A subdivision all of whose elements are simplices is a triangulation.
For subdivisions S , S ′ , we say that S is a refinement of S ′ if every element of S is a subset of an element of S ′ . Refinement gives a poset structure on the set of all subdivisions of A. The maximal element of this poset is the trivial subdivision and the minimal elements are the triangulations.

Flips
As stated in the introduction, two triangulations are connected by a flip if there is a subdivision whose only proper refinements are these two triangulations. We will now give an equivalent definition of a flip which will be easier to use.
A circuit is a minimal affinely dependent subset of R d . If X = {x 1 , . . . , x k } is a circuit, then the elements of X satisfy an affine dependence equation for all i, i λ i = 0, and the equation is unique up to multiplication by a constant. This gives a unique partition X = X + ∪ X − of X given by X + = {x i : λ i > 0} and X − = {x i : λ i < 0}. We will write X = (X + , X − ) to denote a choice of which part we call X + and which we call X − .
A circuit X = (X + , X − ) has exactly two non-trivial subdivisions, which are the following triangulations: Given a subdivision S and a cell C ∈ S , we define the link of C in S as We can now state the definition of a flip, in the form of a proposition.
Proposition 2.2 (Santos [11]). Let T be a triangulation of A. Suppose there is a circuit X = (X + , X − ) contained in A such that 2. All maximal simplices of T + X have the same link L in T . Then the collection is a triangulation of A. We say that T has a flip supported on (X + , X − ), and that T ′ is the result of applying this flip to T .

Regular subdivisions
Let A ⊆ R m be as before. Let ω : A → R be any function. For a cell C ⊆ A, we define the lift of C to be the set C ω ⊂ R m × R given by We call a subset F ⊆ A ω a lower face of A ω if either F is empty or there is a linear functional φ ∈ (R m × R) * such that φ(0, 1) > 0 and F is the set of all points which minimize φ on A ω . Then the collection of all C ⊆ A such that C ω is a lower face of A ω is a subdivision of A. We call this the regular subdivision of A with respect to ω, and denote it by F . Both triangulations of a circuit are regular, as stated below.

The product of two simplices
We now consider ∆ m−1 × ∆ n−1 , the product of two simplices of dimensions m − 1 and n − 1. Following the conventions of the previous section, we will understand ∆ m−1 × ∆ n−1 to mean the set of vertices of ∆ m−1 × ∆ n−1 rather than the polytope itself.
Let ∆ m−1 := {e 1 , . . . , e m } be the standard basis for R m and ∆ n−1 := {f 1 , . . . , f n } be the standard basis for R n . We embed ∆ m−1 × ∆ n−1 in R m × R n by Let G := K m,n be the complete bipartite graph with vertex set ∆ m−1 ∪ ∆ n−1 and edge set {e i f j : i ∈ [m], j ∈ [n]}. We have a bijection (e i , f j ) → e i f j between ∆ m−1 ∪ ∆ n−1 and the edge set of G. For each cell C ⊆ ∆ m−1 × ∆ n−1 , let G(C) be the minimal subgraph of G with edge set {e i f j : (e i , f j ) ∈ C}. Then C is a simplex if and only if G(C) is acyclic, and C is a circuit if and only if G(C) is a cycle. If C is a circuit, then G(C) alternates between edges corresponding to positive and negative elements of the circuit.

Zonotopal tilings
We now define zonotopal tilings in terms of mixed subdivisions. We review the Cayley trick which shows that mixed subdivisions are special cases of subdivisions. The information in this section was developed in [14], [4], and [12].

Mixed subdivisions
Let A 1 , . . . , A n be finite subsets of R m . The Minkowski sum of A 1 , . . . , A n is the set of points In this paper, we want the phrase " A i " to identify a set of points but also retain the information of what A 1 , . . . , A n are. In other words, A i will formally mean an ordered tuple (A 1 , . . . , A n ) but by abuse of notation will also refer to the Minkowski sum.
A mixed cell of A mixed cell is fine if all the B i are simplices and lie in independent affine subspaces. A face of a mixed cell B i is a mixed cell F i of B i such that there exists a linear functional φ ∈ (R m ) * such that for all i, either F i = ∅ or F i is the set of all points which minimize φ on B i .
For mixed subdivisions S , S ′ , we say that S is a refinement of S ′ if every element of S is a mixed cell of an element of S ′ . Refinement gives a poset structure on the set of mixed subdivisions of A i whose minimal elements are the fine mixed subdivisions.

Basic sums and zonotopes
Let A 1 , . . . , A n be as in the previous section. Let ∆ m−1 = {e 1 , . . . , e m } be the standard basis of R m . We will say that

The faces of a basic sum
A i can be described as follows: Given a weak ordering ≤ w of ∆ m−1 , for any S ⊆ ∆ m−1 define min(≤ w , S) to be the set of minimal elements of S with respect to ≤ w . Then B i is a face of A i if and only if there is some weak ordering ≤ w such that for all i, . . , A n be as in the previous section, and assume additionally that |A i | = 2 for all i. Then A i is called a zonotope, and its mixed subdivisions are called zonotopal tilings. A fine zonotopal tiling is called a tight zonotopal tiling.
Then Π m−1 := 1≤i<j≤m A ij is a basic zonotope called the (m − 1)-dimensional permutohedron. Every weak ordering ≤ w of ∆ m−1 induces a different face min(≤ w , A i ) of the permutohedron.

Coherent mixed subdivisions
Let A 1 , . . . , A n be finite subsets of R m . For each i = 1, . . . , n, let ω i : A i → R be a function. For a mixed cell We call a mixed cell F i of ( A i ) ω a lower face of ( A i ) ω if there is a linear functional φ ∈ (R m × R) * such that φ(0, 1) > 0 and for each i, either F i is empty or F i is the set of points which minimize φ on A ω i i . The collection of all B i such that ( B i ) ω is a lower face of ( A i ) ω is a mixed subdivision of A i called the coherent mixed subdivision of A i with respect to ω. We denote it by S ω A i . In the case where A i is basic, we have the following characterization of S ω A i .

Theorem 3.3 (Develin and Sturmfels [3]). Let
A i be basic, and let ω i : A i → R be functions. For any x = (x 1 , . . . , x m ) ∈ R m and A i , let type(x, ω, A i ) be the set of all e j ∈ A i such that Then

The Cayley trick
Let A 1 , . . . , A n be as before. Let ∆ n−1 = {f 1 , . . . , f n } be the standard basis of R n . We define the Cayley embedding of A i to be the following set in R m × R n : The Cayley trick says the following.

C is a bijection between the mixed cells of
A i and the cells of C( A i ), and this map preserves facial relations.

For any mixed subdivision S of
We say that two fine mixed subdivisions T and T ′ are connected by a flip if the triangulations C(T ) and C(T ′ ) are connected by a flip.

A zonotope with disconnected flip graph 4.1 Zonotopal tilings of the 3-permutohedron
We are now ready to construct a zonotope and a nontrivial component of its flip graph. The construction uses the 3-dimensional permutohedron and eight of its zonotopal tilings.
It will be notationally easier to work with Cayley embeddings of zonotopes rather than zonotopes themselves. Thus we will identify a zonotope with its Cayley embedding.
For a set S, let Γ k S denote the set of all ordered k-tuples We now set m = 4. We will construct eight different tilings of Π 3 , each indexed by a different element of Γ 3 4 .
It is easy to check (using Theorem 3.3, for example) that T γ Π 3 is a triangulation. For each γ ′ ∈ Γ 3 4 , the circuit X γ ′ is a face of Π 3 , and thus T γ Π 3 contains the triangulation it induces on X γ ′ . From the definition of ω, we have Thus, by Proposition 2.3, T γ Π 3 induces the following triangulations on the circuits X γ ′ : The key property of T γ Π 3 is that it only has flips on the circuits X (ijl) , X (jkl) , and X (kil) . The idea will be to tile a larger zonotope with 3-permutohedra and then tile each 3permutohedron with some T γ Π 3 so that in the end, no circuit of size six can be flipped. To help with this construction, we will define a group action on Γ 3 4 .
It is easy to check that γ is determined by o γ . Now, we will map each α ∈ Γ 2 4 to a permutation π α : Γ 3 4 → Γ 3 4 . This map is completely determined by the following rules: For any distinct i, j, k, l ∈ [4], we have Let G Γ 3 4 be the permutation group of Γ 3 4 generated by all the π α .
Proposition 4.1. The following are true.

For
Proof. Since each γ is determined by o γ , we can view G Γ 3 4 as an action on the set of functions o γ . We check that for all distinct i, j, k, l ∈ [4] and γ ∈ Γ 3 4 , we have It follows that we can embed G Γ 3 4 as a subgroup of Z 4 2 . This implies that every element of G Γ 3 4 is an involution and G Γ 3 4 is abelian. It is also easy to check from the above action on the o γ that every element of Γ 3 4 has orbit of size 8, and hence G Γ 3 4 is transitive. From the above action on o γ , we see that H l maps Γ 3 4 (ijk) to itself and every element of Γ 3 4 (ijk) has orbit of size 4 under H l . Since |Γ 3 4 (ijk)| = 4, Γ 3 4 (ijk) is an orbit of H l .

A zonotope and a component of its flip graph
Let N be a positive integer to be determined later. For each distinct i, j ∈ [4] and −N ≤ r ≤ N , we create a variable f r ij , and we make the identification of variables be the 3-dimensional permutohedron with 2N + 1 copies of each generating vector. We will now prove our first main result. For distinct i, j, k ∈ [4] and for any −N ≤ r, s, t ≤ N , let X rst ijk = ((X rst ijk ) + , (X rst ijk ) − ) be the circuit with affine dependence relation Lemma identifies a component of the flip graph of Π.
Let S C be the set of all triangulations of Π which contain every element of C as a subset. Then S C is closed under flips.
Proof. The proof follows immediately from the following two facts, both of which are proved in the appendix.
Proposition 4.4. Let T be a triangulation of Π, and suppose that there are distinct i, j, k, l ∈ [4] and 1 ≤ r, s, t, u, v, w ≤ N such that Then T does not have a flip supported on X rst ijk . Proposition 4.5. Let T be a triangulation of Π such that T rst ijk ⊆ T . Let T ′ be the result of a flip on T which is not supported on X rst ijk . Then T rst ijk ⊆ T ′ .
It remains to prove that there is some C for which S C is neither empty nor the whole set of triangulations of Π. This will be done in the next section.

Construction of C and some T ∈ S C
We first consider the regular subdivision S ω Π where ω : Π → R is a function such that 4] and −N ≤ r ≤ N . The cells of S ω Π are described as follows.
Proposition 4.6. Let X be the set of x = (x 1 , x 2 , x 3 , x 4 ) ∈ R 4 such that x 1 + · · · + x 4 = 0 and if ijkl is a permutation of [4] such that x i ≥ x j ≥ x k ≥ x l , then x i − x j , x j − x k , and x k − x l are integers at most N . Let X * be the set of x ∈ X such that |x i − x j | ≤ N for all i, j ∈ [n]. The following are true.
2. If x ∈ X * , then C(x) is the Cayley embedding of a translated 3-permutohedron.
and D is a simplex affinely independent to Π(x).
3. Let x ∈ X \ X * . Suppose F 1 , . . . , F k are faces of C(x) and T 1 , . . . , T k are triangulations of these faces, respectively, which agree on intersections of these faces. Then there is a triangulation of C(x) which contains T 1 , . . . , T k as subsets.
Proof. Part 1 follows from Theorem 3.3. Part 2 immediately follows. The only nontrivial case of part 3 is when x satisfies x i ≥ x j ≥ x k ≥ x l for some permutation ijkl of [4], In this case C(x) is of the form X rst ijk ∪X suv jkl ∪D, where D is a simplex affinely independent to X rst ijk ∪ X suv jkl . By Proposition 5.10 (or by an easy check), any triangulations of X rst ijk and X suv jkl can be extended to a triangulation of X rst ijk ∪ X suv jkl , and hence to a triangulation of C(x).
For each x ∈ X * , we have an affine isomorphism Π 3 → Π(x) given by f (ij) → f be the image of T γ Π 3 under this isomorphism. We will now choose a random triangulation of every C(x), x ∈ X * , as follows: 1. For each 1 ≤ i < j ≤ 4 and −N ≤ r ≤ N , let g r ij = g −r ji be an independent random element of G Γ 3 4 which is 1 with probability 1/2 and π (ij) with probability 1/2.

For each
Proposition 4.7. For any two x, x ′ ∈ X * , the triangulations T C(x) and T C(x ′ ) agree on the common face of C(x) and C(x ′ ).
Proof. The only non-trivial case is when C(x) ∩ C(x ′ ) contains a circuit X rst ijk . We need to check that T C(x) and T C(x ′ ) agree on this circuit. If X rst ijk ⊆ C(x) ∩ C(x ′ ), then On the other hand, by Proposition 4.1 (2), o γ(x) ({i, j, k}) depends only on g Π(x ′ ) contain the same triangulation of X rst ijk , as desired.
By Proposition 4.7 and Proposition 4.6(3), we can thus extend the above triangulations of the C(x) to a triangulation T of Π. Let C be the collection of all triangulations T rst ijk ⊆ T with i, j, k ∈ [4] distinct, −N ≤ r, s, t ≤ N , and r + s + t = 0. We prove that C satisfies the hypotheses of Lemma 4.3. We will actually prove the following stronger statement, which we will need in the next section. (B) For every T rst ijk ∈ C and γ ∈ Γ 3 4 (ijk), there exists x ∈ X * such that X rst ijk ⊆ Π(x) and T [Π(x)] = T γ Π(x) . In particular, when γ = (ijk), this implies there is some Proof. First, note that for any distinct i, j, k ∈ [4] and −N ≤ r, s, t ≤ N with r + s + t = 0, there is some x ∈ X * such that X rst ijk ⊆ Π(x). Hence C contains a triangulation of X rst ijk . Letting γ = g r ij g s jk g t ki (123), we have By the union bound, the probability that this happens for some distinct i, j, k ∈ [4] and −N ≤ r ≤ N is at most This gives an upper bound on the probability of (A) not happening. We now do the same for (B). Fix T rst ijk ∈ C and γ ∈ Γ 3 4 (ijk). Let H be the set of all x ∈ X * such that X rst ijk ⊆ Π(x). Note that |H| ≥ N . Let {l} = [4] \ {i, j, k}. We have T rst ijk ∈ C , which happens if and only if g r ij g s jk g t ki (123) ∈ Γ 3 4 (ijk). Suppose we fix g r ij , g s jk , and g t ki such that g r ij g s jk g t ki (123) ∈ Γ 3 4 (ijk). Then for each x ∈ H, it follows from Proposition 4.1 and the definition of γ(x) that γ(x) = γ with probability 1/4. Moreover, since (x i − x l , x j − x l , x k − x l ) is different for each x ∈ H, these probabilities are mutually independent for all x ∈ H. Now, for each x ∈ X * we have T [Π(x)] = T γ(x) Π(x) . Thus the probability that there is no The probability that this occurs for some distinct i, j, k ∈ [4], −N ≤ r, s, t ≤ N with r + s + t = 0, and γ ∈ Γ Thus there exists C which satisfies the hypotheses of Lemma 4.3 and T ∈ S C . There are triangulations of Π which are not in S C ; for example, a different choice of the g r ij would yield a triangulation which does not contain every element of C . This proves Theorem 4.2.

A product of two simplices with disconnected flip graph
We will use the construction from the previous section to show that the product of two simplices does not in general have connected flip graph. The idea will be to go up one dimension and construct multiple copies of T in different directions in this space.
Lemma 5.2. For each S ∈ [5] 4 and distinct i, j ∈ S, let C S,i,j be a collection of zonotopal triangulations of the form Assume that for each S ∈ [5] 4 and distinct i, j ∈ S, and for each T f 1 f 2 f 3 i 1 i 2 i 3 ∈ C S,i,j , the following are true.

If {i
In addition, assume the following about the C S,i,j .
Then S C is closed under flips.
Proof. We need the following three general facts about triangulations of A. The first is true for all point sets A and is proved in [7]. The other two are proved in the Appendix. Proposition 5.3. Let T be a triangulation of A and let X = (X + , X − ) be a circuit in A. Suppose that X − ∈ T . Then T has a flip supported on (X + , X − ) if and only if there is no maximal simplex τ ∈ T with X − ⊆ τ and |X ∩ τ | ≤ |X| − 2.
Proposition 5.4. Let T be a triangulation of A. Let i, j, k, l ∈ [5] be distinct, and for each α ∈ Γ 2 {i,j,k,l} , let f α ∈ ∆ n−1 . Suppose that Then T does not have a flip supported on X Proposition 5.5. Let T be a triangulation of A and let T ′ be the result of a flip on T supported on X = (X + , X − ). Suppose that σ ∈ T and σ / ∈ T ′ , and G(σ) is connected. Then σ contains a maximal simplex of T + X . We also need the following two facts about flips on elements of S C . Proposition 5.6. Let i, j, l ∈ [5] be distinct and let f 1 ∈ ∆ (ij) , f 2 ∈ ∆ (il) . Let T ∈ S C . Then T does not have a flip supported on X f 1 f 2 ij .
Proof. By Property 3 of C S,i,j , there exist k ∈ [5] \ {i, j, l}, f ′ 2 ∈ ∆ (jk) , and f ′ 3 ∈ ∆ (ki) such that T If τ is a maximal simplex in T containing σ, then ( So by Proposition 5.3, T does not have a flip supported on X f 1 f 2 ij .
Proof. This is a direct corollary of Property 1 of C S,i,j and Proposition 5.4.
We now proceed with the proof. Suppose that T ∈ S C , and let T ′ be the result of a flip on T supported on X = (X + , X − ). We prove that properties (i) and (ii) hold for T ′ .
X . This leaves two cases for X.
1. Case 1: X has size 4. Thus we can write X = X However, this contradicts Proposition 5.6. So we cannot have |X| = 4.

Case 2:
This contradicts Proposition 5.7. Hence we must have with variables as defined in (ii). By Proposition 5.5, σ contains a maximal simplex of X. By the same argument as in part (i), we cannot have |X| = 4 or X = X f 1 f 2 f 3 ijk . This leaves as the only possibility.
We show that T cannot have a flip on this circuit. Let l ′ = S \ {i, j, k}. By Property 2 By Property (ii), the first of these inclusions implies that and hence X This inclusion along with the last two inclusions of (5.1) imply, by Proposition 5.4, that T does not have a flip on X f f 3 f 2 jik , as desired. In particular, if T can be extended to a triangulation T ′ of A, then we will have some T ′ ∈ S C . The next Proposition guarantees we can always do this.

Reduction to zonotopes
Proposition 5.8. If T is a triangulation of Π, then there is a triangulation T ′ of A with T ⊆ T ′ .
Proof. Let S A := S ω A be the regular subdivision of A with ω : A → R defined as follows. For each distinct i, j, k ∈ [5] and f ∈ ∆ (ij) , let ǫ f,k > 0 be a generic positive real number, and define ω(e i , f ) = 0 ω(e j , f ) = 0 ω(e k , f ) = ǫ f,k .
Then S A contains Π as a cell, corresponding to x = 0 under the notation of Theorem 3.3.
Since the ǫ f,k are generic, all other cells of S A can be written as F ∪ D, where F is a face of Π and D is a simplex affinely independent to F . Hence, any triangulation of the cell Π can be extended to a refinement of S A which is a triangulation of A.

Construction of a zonotopal tiling
We now construct a triangulation of Π from which we will obtain our collections C S,i,j . For each (ij) ∈ Γ 2 5 , partition ∆ (ij) into the following sets: ∆ i,j , ∆ j,i , and Next, for each distinct i, j ∈ [5], choose an element f * i,j ∈ ∆ i,j . For each S ∈ [5] 4 with i, j ∈ S, let ∆ S,i,j,(ij) ⊆ ∆ i,j be sets such that The sizes of all of these sets will be determined later. Now, we let S := S ω Π be the regular subdivision of Π with ω : Π → R defined as follows. First, for every distinct i, j ∈ [5], we set Finally, for each S ∈ [5] 4 and distinct i, j, k ∈ S, let 0 < ǫ S,i,j,k < 1 be a generic real number. We set We analyze the cells of S . For each S ∈ [5] 4 and distinct i, j ∈ S, let Π S,i,j := In addition, for each k ∈ S \ {i, j}, let Proposition 5.9. Every cell of S is of the form C ∪ D, where D is a simplex affinely independent to C and C is a face of one of the following.
We will call the cells in (a) and (b) the complex cells of S . Proof. Using the notation of Theorem 3.3, the cell in (a) corresponds to x ∈ R 5 with x i = 0, x j = 1, x l = ǫ S,i,j,l for each l ∈ S \ {i, j}, and x k = ǫ S ′ ,i,j,k . The cell in (b) corresponds to x ∈ R 5 with x i = 0, x j = 1, x k = ǫ S,i,j,k , x k ′ = ǫ S ′ ,i,j,k ′ , and x k ′′ = ǫ S ′′ ,i,j,k ′′ . Due to the genericness of the ǫ S,i,j,k , it can be checked that every cell of S is the union of a face of one of these cells and an affinely independent simplex; we leave the details to the reader.
Thus, in order to give a triangulation of Π which refines S , it suffices to specify triangulations of the complex cells of S which agree on common faces. To do this, we first specify triangulations of each Π S,i,j . We will then use a "pseudoproduct" operation to extend these to triangulations of the complex cells.
Fix S ∈ [5] 4 and distinct i, j ∈ S. LetΠ be the large 3-permutohedron defined in equation (4.2). Let ψ : [4] → S be a map such that ψ(1) = i and ψ(2) = j. We now choose the sizes of the ∆ S,i,j,(i ′ j ′ ) so that we have an affine isomorphism Ψ :Π → Π S,i,j given by e k → e ψ(k) for all k ∈ [4] and so that {f r kl } −N ≤r≤N maps bijectively to ∆ S,i,j,(ψ(k)ψ(l)) . We will define Ψ :Π → Π S,i,j so that f −N 12 maps to f * i,j . LetT andC be the triangulation ofΠ and the collection of triangulations, respectively, constructed in Section 4.4 which satisfy Proposition 4.8. Let T S,i,j andC S,i,j be the images ofT andC , respectively, under Ψ. We thus have a triangulation T S,i,j of each Π S,i,j .
To extend the T S,i,j to triangulations of the complex cells, we use the following "ordered pseudoproduct" construction. It is proved in the Appendix.
Then M is the set of maximal simplices of a triangulation T (T 1 , · · · , T N ) of Π 1 ∪ · · ·∪ Π N , and T r ⊆ T (T 1 , · · · , T N ) for all 1 ≤ r ≤ N . Now, fix distinct i, j ∈ [5], and specify an ordering S 1 , S 2 , S 3 of the sets S ∈ [5] 4 which contain i and j. Each complex cell is of the form By Proposition 5.10, we thus have a triangulation T ( We leave it as an exercise to verify that all of these triangulations of the complex cells agree on common faces. 1 Hence, we can extend these triangulations of the complex cells to a triangulation T of Π. By Proposition 5.10, this triangulation T contains all the triangulations T S,i,j as subcollections.
For each S ∈ [5] 4 and distinct i, j ∈ [n], let D S,i,j be the set of all zonotopal triangulations of the form T f 1 f 2 f 3 ijk such that k ∈ S \ {i, j} and T We now finally show that the assumptions of Lemma 5.2 hold for C S,i,j and T .
Proposition 5.11. The collections C S,i,j satisfy Properties 1-3 of Lemma 5.2. 1 The argument is the same as the proof of Property 1 in the proof Proposition 5.10.

Proof. Let
By considering a maximal simplex of T containing σ 0 , we have σ 0 ∪ {(e, f 1 )} ∈ T for some e ∈ {e i , e j }. If e = e i , then {(e i , f 1 ), (e j , f * i,j )} ⊆ σ 0 . However, this contradicts Proposition 5.13 and Proposition 6.3 in the Appendix. Thus e = e j . Hence, we have Hence, we have collections C S,i,j which satisfy the hypotheses of Lemma 5.2, and by Propositions 5.12 and 5.8, there exists a triangulation T ′ ∈ S C . Clearly S C is not the set of all triangulations of A; for example, there exist triangulations of A which do not contain any triangulations of circuits of size six [5]. Hence Lemma 5.2 proves Theorem 5.1. We prove the following general fact. Proposition 6.1. Let A be a subset of ∆ m−1 × ∆ n−1 , and let T be a triangulation of A. Let T ′ be the result of a flip on T supported on X = (X + , X − ). Suppose that σ ∈ T and σ / ∈ T ′ , and G(σ) is connected. Then σ contains a maximal simplex of T + X . Proof. Let τ be a maximal simplex of T containing σ. We must have τ / ∈ T ′ since σ / ∈ T ′ . So τ must contain a maximal simplex τ ′ of T + X . Moreover, since τ \ {x} ∈ T ′ for any x ∈ X − , we must have σ ⊆ τ \ {x} for any x ∈ X − . Hence σ ⊇ X − . Since G(σ) is connected and G(σ ∪ τ ′ ) ⊆ G(τ ) is acyclic, this can only happen σ ⊇ τ ′ , as desired.
Proposition 5.5 now follows immediately. Proposition 4.5 follows by applying Proposition 6.1 to each maximal simplex σ of T rst ijk and noting that σ cannot contain a maximal simplex of a circuit X of Π except when X = X rst ijk . This is because for a zonotope A, each element of ∆ n−1 has only two neighbors in G(A).
Proof of Property 2. Let σ = r<s (σ r \ ρ) ∪ σ s ∪ r>s σ r be an element of M as before. Let x ∈ σ, and consider the facet σ \ {x}. We have the following cases.
Case 2: x ∈ ρ. Subcase 2.1: σ ∩ ρ = ρ. Then σ \ {x} is contained in a facet of Π 1 ∪ · · · ∪ Π N because f has no neighbors in G(σ \ {x}). Subcase 2.2: σ ∩ ρ = ρ. This implies s = N and x ∈ σ r for all r. First, suppose there is some t such that σ t \ {x} is not contained in a facet of Π t , and let t be the largest such number. Then there is some x ′ ∈ Π t such that σ ′ t := σ t \ {x} ∪ {x ′ } ∈ T t . Thus is an element of M containing σ \ {x}, as desired. 4 Now suppose there is no such t. Then for all r, σ r \ {x} is contained in a facet F r of Π r . We have x / ∈ F r since σ r is full dimensional in Π r . For each r, let φ r : Π r → R be a linear functional supporting F r on Π r . Let {y} = ρ \ {x}. Then y ∈ F r and x / ∈ F r , so φ r (y) < φ r (x). By appropriately choosing φ r , we may assume φ r (y) = 0 and φ r (x) = 1 for all r. As before, we can then define a linear functional φ ′ on Π 1 ∪ · · · ∪ Π N such that φ ′ (v) = φ r (v) for all v ∈ Π r . Then the face of Π 1 ∪ · · · ∪ Π N supported by φ ′ contains σ \ {x}, and this face is proper. Thus σ \ {x} is contained in a facet of Π 1 ∪ · · · ∪ Π N .
Case 3: x ∈ σ s \ ρ. Subcase 3.1: σ s \ {x} is not contained in a facet of Π s . Then there is some x ′ ∈ Π s such that σ ′ s := σ s \ {x} ∪ {x ′ } ∈ T s . First assume that x ′ / ∈ ρ. Then is an element of M containing σ \ {x}, as desired. Now assume that x ′ ∈ ρ. First, suppose that for all r > s, σ ′ r := σ r ∪ {x ′ } ∈ T r . Then is an element of M containing σ \ {x}, as desired. Now suppose that there is some t > s such that σ t ∪ {x ′ } / ∈ T t , and let t be the smallest such number. Let x ′′ be a point in T t such that x ′′ / ∈ σ t and σ ′′ t := σ t ∪ {x ′′ } ∈ T t . By definition of t, x ′′ = x ′ , and hence x ′′ / ∈ ρ. Then is an element of M containing σ \ {x}, as desired. Subcase 3.2: σ s \ {x} is contained in a facet F of Π s . If ρ ⊆ F , then by the same argument as in Subcase 1.2, σ \ {x} is contained in a facet of Π 1 ∪ · · · ∪ Π N . So we may assume ρ ⊆ F . In particular, this means σ s ∩ ρ = ρ \ {x ′ } for some x ′ ∈ ρ.
First, suppose that there is some t < s such that σ t \ {x ′ } is not contained in a facet of T t , and let t be the largest such number. Then there is some x ′′ ∈ T t such that is an element of M containing σ \ {x}, as desired.