A Zonotope and a Product of Two Simplices with Disconnected Flip Graphs

Abstract

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.

1 Introduction

We consider the poset $\mathcal P$ 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 $\mathcal P$ are connected by a flip if there is an element of $\mathcal P$ 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 $\mathcal P$ and whose edges are flips.

We are concerned in this paper with connectivity of the flip graph. For zonotopal tilings, the flip graph is known to be connected for cyclic zonotopes [19] or if the zonotope has dimension two [5]. Our first result is the first example 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 the first example of a polytope whose flip graph of triangulations is not connected. However, the case when P is a product of two simplices remained of special interest due to the appearances of these triangulations in various branches of mathematics; see [2, Chap. 6.2] for an overview. Santos [14] proved that the flip graph of $\Delta ^2 \times \Delta ^n$ is connected for all n, and the author [6] proved that the flip graph of $\Delta ^3 \times \Delta ^n$ is connected for all n. However, we show that the flip graph of $\Delta ^4 \times \Delta ^n$ is not connected for $n \approx 4 \cdot 10^4$.

Triangulations of lattice polytopes are closely related to toric varieties in algebraic geometry. Each lattice polytope defines a toric ideal and an associated toric Hilbert scheme. For a totally unimodular polytope, the associated toric Hilbert scheme is connected if and only if the flip graph of the polytope is connected; see [17, Chap. 10] or [7]. Thus, our result implies that the toric Hilbert scheme of $\Delta ^{4} \times \Delta ^{n}$ is not connected for large n. While non-connected toric Hilbert schemes had previously been constructed [13], our proof demonstrates non-connectivity for the first time for a totally unimodular polytope. In addition, the toric ideal associated to $\Delta ^m \times \Delta ^n$ is the well-studied determinantal ideal generated by $2 \times 2$ minors of an $(m+1) \times (n+1)$ matrix; its zero-locus is the Segre variety.

Connectivity of flip graphs is also related to the generalized Baues problem, formulated by Billera et al. [1], which concerns the topology of the Baues poset $\mathcal P$. Specifically, the problem asks if the order complex of $\mathcal P$ minus its maximal element is homotopy equivalent to a sphere. While the problem has been resolved for most cases of interest, it remains open for zonotopes, and is of particular interest in this case because it is equivalent to the extension space conjecture for oriented matroids, which states that the extension space of a realizable oriented matroid is homotopy equivalent to a sphere. See [11, 18] for definitions and known results. In general, the generalized Baues problem and the question of flip graph connectivity 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 [15] by constructing a point set in general position with disconnected flip graph. The problem for zonotopal tilings remains open, but would be answered in the negative if a zonotope with generating vectors in general position was found to have disconnected flip graph; see [9, Lem. 3.1].

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 $\Delta ^4 \times \Delta ^n$ is not connected. Section 6 is an appendix proving several propositions used in the paper.

2 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. [2] for a more comprehensive treatment.

2.1 Subdivisions and Triangulations

Throughout this section, let $A \subset \mathbb R^m$ 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 $F \subseteq C$ such that there exists a linear functional $\phi \in (\mathbb R^m)^*$ such that F is the set of all points which minimize $\phi$ on C. For any cell C, let ${{\mathrm{conv}}}(C)$ denote the convex hull of C.

Definition 2.1

A polyhedral subdivision, or subdivision, of A is a collection $\mathscr {S}$ of cells of A such that

• 1. If $C \in \mathscr {S}$ and F is a face of C, then $F \in \mathscr {S}$.

• 2. If $C,C' \in \mathscr {S}$, then ${{\mathrm{conv}}}(C) \cap {{\mathrm{conv}}}(C') = {{\mathrm{conv}}}(F)$ where F is a face of C and $C'$.

• 3. $\bigcup _{C \in \mathscr {S}} {{\mathrm{conv}}}(C) = {{\mathrm{conv}}}(A)$.

The subdivision consisting of A and all faces of A is the trivial subdivision. A subdivision all of whose elements are simplices is a triangulation.

If $\mathscr {S}$ is a subdivision of A and F is a face of A, then $\mathscr {S}$ induces a subdivision $\mathscr {S}[F]$ of F by

$$\begin{aligned} \mathscr {S}[F] := \{ C \in \mathscr {S} : C \subseteq F \}. \end{aligned}$$

For subdivisions $\mathscr {S},\mathscr {S}'$, we say that $\mathscr {S}$ is a refinement of $\mathscr {S}'$ if every element of $\mathscr {S}$ is a subset of an element of $\mathscr {S}'$. 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.

2.2 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 $\mathbb R^m$. If $X = \{x_1,\cdots ,x_k\}$ is a circuit, then the elements of X satisfy an affine dependence equation

$$\begin{aligned} \sum _{i=1}^k \lambda _i x_i = 0, \end{aligned}$$

where $\lambda _i \in \mathbb R \,\backslash \,\{0\}$ for all i, $\sum _i \lambda _i = 0$, and the equation is unique up to multiplication by a constant. This gives a unique partition $X = X^+ \cup X^-$ of X given by $X^+ = \{x_i : \lambda _i > 0\}$ and $X^- = \{x_i : \lambda _i < 0\}$. 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:

$$\begin{aligned} \mathscr {T}_X^+ \,{:=}\, \{ \sigma \subseteq X : \sigma \not \supseteq X^+ \}, \qquad \mathscr {T}_X^- \,{:=}\, \{ \sigma \subseteq X : \sigma \not \supseteq X^- \}. \end{aligned}$$

Given a subdivision $\mathscr {S}$ and a cell $C \in \mathscr {S}$, we define the link of C in $\mathscr {S}$ as

$$\begin{aligned} {{\mathrm{link}}}_{\mathscr {S}}(C) \,{:=}\, \{ C' \in \mathscr {S} : C \cap C' = \emptyset ,\, C \cup C' \in \mathscr {S} \}. \end{aligned}$$

We can now state the definition of a flip, in the form of a proposition.

Proposition 2.2

(Santos [12]) Let $\mathscr {T}$ be a triangulation of A. Suppose there is a circuit $X = (X^+,X^-)$ contained in A such that

• 1. $\mathscr {T}_X^+ \subseteq \mathscr {T}$.

• 2. All maximal simplices of $\mathscr {T}_X^+$ have the same link $\mathscr {L}$ in $\mathscr {T}$.

Then the collection

$$\begin{aligned} \mathscr {T}' \,{:=}\, \mathscr {T} \,\backslash \,\{ \rho \cup \sigma : \rho \in \mathscr {L}, \sigma \in \mathscr {T}_X^+ \} \cup \{ \rho \cup \sigma : \rho \in \mathscr {L}, \sigma \in \mathscr {T}_X^- \} \end{aligned}$$

is a triangulation of A. We say that $\mathscr {T}$ has a flip supported on $(X^+,X^-)$, and that $\mathscr {T}'$ is the result of applying this flip to $\mathscr {T}$. The flip graph (on triangulations) of $\mathscr {T}$ is the graph whose vertices are triangulations of $\mathscr {T}$ and with an edge between two triangulations if one is obtained from the other by a flip.

The following fact is a convenient way to determine whether a flip supported on a certain circuit exists. It is proved in [6].

Proposition 2.3

Let $\mathscr {T}$ be a triangulation of A and let $X = (X^+,X^-)$ be a circuit in A. Suppose that $X^- \in \mathscr {T}$. Then $\mathscr {T}$ has a flip supported on $(X^+,X^-)$ if and only if there is no maximal simplex $\tau \in \mathscr {T}$ with $X^- \subseteq \tau$ and $|X \cap \tau | \le |X | - 2$.

2.3 Regular Subdivisions

Let $A \subseteq \mathbb R^m$ be as before. Let $\omega :A \rightarrow \mathbb R$ be any function. For a cell $C \subseteq A$, we define the lift of C to be the set $C^\omega \subset \mathbb R^m\!\times \!\mathbb R$ given by

$$\begin{aligned} C^\omega \,{:=}\, \{ (x, \omega (x)) : x \in C \}. \end{aligned}$$

We call a subset $F \subseteq A^\omega$ a lower face of $A^\omega$ if either F is empty or there is a linear functional $\phi \in (\mathbb R^m \times \mathbb R)^*$ such that $\phi (0,1) > 0$ and F is the set of all points which minimize $\phi$ on $A^\omega$. Then the collection of all $C \subseteq A$ such that $C^\omega$ is a lower face of $A^\omega$ is a subdivision of A. We call this the regular subdivision of A with respect to $\omega$, and denote it by $\mathscr {S}_A^\omega$.

If F is a face of A, then $\mathscr {S}_A^\omega [F] = \mathscr {S}_F^{\omega \vert _F}$.

Both triangulations of a circuit are regular, as stated below.

Proposition 2.4

Suppose $\sum _{i=1}^k \lambda _i x_i = 0$ is the affine dependence equation for a circuit $X = (X^+,X^-)$ with $X^+ = \{x_i : \lambda _i > 0\}$ and $X^- = \{x_i : \lambda _i < 0\}$. Let $\omega :X \rightarrow \mathbb R$ be a function. Then

$$\begin{aligned} \mathscr {S}_X^\omega = {\left\{ \begin{array}{ll} \mathscr {T}_X^+ &{} \mathrm{if} \ \sum _{i=1}^k \lambda _i \omega (x_i) > 0, \\ \mathscr {T}_X^- &{} \mathrm{if} \ \sum _{i=1}^k \lambda _i \omega (x_i) < 0. \end{array}\right. } \end{aligned}$$

2.4 The Product of Two Simplices

We now consider $\Delta ^{m-1} \times \Delta ^{n-1}$, the product of two simplices of dimensions $m-1$ and $n-1$. Following the conventions of the previous section, we will understand $\Delta ^{m-1} \times \Delta ^{n-1}$ to mean the set of vertices of $\Delta ^{m-1} \times \Delta ^{n-1}$ rather than the polytope itself.

Let $\Delta ^{m-1} \,{:=}\, \{e_1, \cdots , e_m\}$ be the standard basis for $\mathbb R^m$ and $\Delta ^{n-1} \,{:=}\, \{f_1, \cdots , f_n\}$ be the standard basis for $\mathbb R^n$. We embed $\Delta ^{m-1} \times \Delta ^{n-1}$ in $\mathbb R^m \times \mathbb R^n$ by

$$\begin{aligned} \Delta ^{m-1} \times \Delta ^{n-1} \,{:=}\, \{ (e_i, f_j) : i \in [m], j \in [n] \}. \end{aligned}$$

Let $G := K_{m,n}$ be the complete bipartite graph with vertex set $\Delta ^{m-1} \cup \Delta ^{n-1}$ and edge set $\{ e_if_j : i \in [m], j \in [n]\}$. We have a bijection $(e_i,f_j) \mapsto e_if_j$ between $\Delta ^{m-1} \cup \Delta ^{n-1}$ and the edge set of G. For each cell $C \subseteq \Delta ^{m-1} \times \Delta ^{n-1}$, let G(C) be the minimal subgraph of G with edge set $\{ e_if_j : (e_i,f_j) \in C\}$. 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.

3 Zonotopal Tilings

We now define zonotopal tilings in terms of mixed subdivisions. We review the Cayley trick which shows that mixed subdivisions can be thought of as polyhedral subdivisions of another polytope. The information in this section was developed in [4, 14], and [16].

3.1 Mixed Subdivisions

Let $A_1, \dots , A_n$ be finite subsets of $\mathbb R^m$. The Minkowski sum of $A_1, \dots , A_n$ is the set of points

$$\begin{aligned} \sum A_i = A_1 + \cdots + A_n \,{:=}\, \{ x_1 + \cdots + x_n : x_i \in A_i \text { for all } i \}. \end{aligned}$$

In this paper, we want the phrase “$\sum A_i$” to identify a set of points but also retain the information of what $A_1, \dots , A_n$ are. In other words, $\sum A_i$ will formally mean an ordered tuple $(A_1,\dots ,A_n)$ but by abuse of notation will also refer to the Minkowski sum.

A mixed cell of $\sum A_i$ is a set $\sum B_i$ where $B_i$ is a cell of $A_i$ for all i. A mixed cell is fine if all the $B_i$ are simplices and lie in independent affine subspaces. A face of a mixed cell $\sum B_i$ is a mixed cell $\sum F_i$ of $\sum B_i$ such that there exists a linear functional $\phi \in (\mathbb R^m)^*$ such that for all i, either $F_i = \emptyset$ or $F_i$ is the set of all points which minimize $\phi$ on $B_i$.

Definition 3.1

A mixed subdivision of $\sum A_i$ is a collection $\mathscr {S}$ of mixed cells of $\sum A_i$ such that

• 1. If $\sum B_i \in \mathscr {S}$ and $\sum F_i$ is a face of $\sum B_i$, then $\sum F_i \in \mathscr {S}$.

• 2. If $\sum B_i,\sum B_i' \in \mathscr {S}$, then ${{\mathrm{conv}}}\bigl (\sum B_i\bigr ) \cap {{\mathrm{conv}}}\bigl (\sum B_i'\bigr ) = {{\mathrm{conv}}}\bigl (\sum F_i\bigr )$ where $\sum F_i$ is a face of $\sum B_i$ and $\sum B_i'$.

• 3. $\bigcup _{\sum B_i \in \mathscr {S}} {{\mathrm{conv}}}\bigl (\sum B_i\bigr ) = {{\mathrm{conv}}}\bigl (\sum A_i\bigr )$.

A mixed subdivision is fine if all of its elements are fine.

For mixed subdivisions $\mathscr {S},\mathscr {S}'$, we say that $\mathscr {S}$ is a refinement of $\mathscr {S}'$ if every element of $\mathscr {S}$ is a mixed cell of an element of $\mathscr {S}'$. Refinement gives a poset structure on the set of mixed subdivisions of $\sum A_i$ whose minimal elements are the fine mixed subdivisions.

3.2 Basic Sums and Zonotopes

Let $A_1, \dots , A_n$ be as in the previous section. Let $\Delta ^{m-1} = \{e_1,\dots ,e_m\}$ be the standard basis of $\mathbb R^m$. We will say that $\sum A_i$ is basic if $A_i \subseteq \Delta ^{m-1}$ for all i.

Let $A_1, \dots , A_n$ be as in the previous section, and assume additionally that $|A_i | = 2$ for all i. Then $\sum A_i$ is called a zonotope, and its mixed subdivisions are called zonotopal tilings. A fine zonotopal tiling is called a tight zonotopal tiling.

Example 3.2

For all $1 \le i < j \le m$, let $A_{ij} = \{e_i,e_j\}$. Then $\Pi ^{m-1} \,{:=} \sum _{1 \le i < j \le m} A_{ij}$ is a basic zonotope called the $(m-1)$-dimensional permutohedron.

3.3 Coherent Mixed Subdivisions

Let $A_1, \dots , A_n$ be finite subsets of $\mathbb R^m$. For each $i = 1, \dots , n$, let $\omega _i :A_i \rightarrow \mathbb R$ be a function. For a mixed cell $\sum B_i$ of $\sum A_i$, define the lift $\bigl (\sum B_i\bigr )^\omega$ by

$$\begin{aligned} \left( \sum B_i \right) ^\omega {:=} \sum B_i^{\omega _i}. \end{aligned}$$

We call a mixed cell $\sum F_i$ of $\bigl (\sum A_i\bigr )^\omega$ a lower face of $\bigl (\sum A_i\bigr )^\omega$ if there is a linear functional $\phi \in (\mathbb R^m \times \mathbb R)^*$ such that $\phi (0,1) > 0$ and for each i, either $F_i$ is empty or $F_i$ is the set of points which minimize $\phi$ on $A_i^{\omega _i}$. The collection of all $\sum B_i$ such that $\bigl (\sum B_i\bigr )^\omega$ is a lower face of $\bigl (\sum A_i\bigr )^\omega$ is a mixed subdivision of $\sum A_i$ called the coherent mixed subdivision of $\sum A_i$ with respect to $\omega$. We denote it by $\mathscr {S}_{\sum A_i}^\omega$.

In the case where $\sum A_i$ is basic, we have the following characterization of $\mathscr {S}_{\sum A_i}^\omega$.

Theorem 3.3

(Develin and Sturmfels [3]) Let $\sum A_i$ be basic, and let $\omega _i :A_i \rightarrow \mathbb R$ be functions. For any $x = (x_1,\cdots ,x_m) \in \mathbb R^m$ and $A_i$, let ${{\mathrm{type}}}(x,\omega ,A_i)$ be the set of all $e_j \in A_i$ such that

$$\begin{aligned} x_j - \omega _i(e_j) = \max {\{ x_k - \omega _i(e_k) : e_k \in A_i \}}. \end{aligned}$$

Then $\sum B_i$ is an element of $\mathscr {S}_{\sum A_i}^\omega$ if and only if there is some $x \in \mathbb R^m$ such that for all i, either $B_i = \emptyset$ or $B_i = {{\mathrm{type}}}(x,\omega ,A_i)$.

3.4 The Cayley Trick

Let $A_1, \dots , A_n$ be as before. Let $\Delta ^{n-1} = \{f_1,\dots ,f_n\}$ be the standard basis of $\mathbb R^n$. We define the Cayley embedding of $\sum A_i$ to be the following set in $\mathbb R^m \times \mathbb R^n$:

$$\begin{aligned} \mathcal C\left( \sum A_i \right) {:=} \bigcup _{i=1}^n \{ (x,f_i) : x \in A_i \}. \end{aligned}$$

The Cayley trick says the following.

Theorem 3.4

(Sturmfels [16], Huber et al. [4]) The following are true.

• 1. $\mathcal C$ is a bijection between the mixed cells of $\sum A_i$ and the cells of $\mathcal C\bigl (\sum A_i\bigr )$, and this map preserves facial relations.

• 2. For any mixed subdivision $\mathscr {S}$ of $\sum A_i$, the collection $\mathcal C(\mathscr {S})$ is a subdivision of $\mathcal C\bigl (\sum A_i\bigr )$. The map $\mathscr {S} \mapsto \mathcal C(\mathscr {S})$ is a poset isomorphism between the mixed subdivisions of $\sum A_i$ and the subdivisions of $\mathcal C\bigl (\sum A_i\bigr )$.

• 3. If $\omega _i :A_i \rightarrow \mathbb R$ are functions for $i = 1, \dots , n$ and $\mathcal C(\omega ) :\mathcal C\bigl (\sum A_i\bigr ) \rightarrow \mathbb R$ is defined as $\mathcal C(\omega )(x,f_i) = \omega _i(x)$, then

  $$\begin{aligned} \mathcal C \Bigl (\mathscr {S}_{\sum A_i}^\omega \Bigr ) = \mathscr {S}_{\,\mathcal C(\sum A_i)}^{\mathcal C(\omega )}. \end{aligned}$$

Example 3.5

If $\sum A_i$ is basic, then $\mathcal C\bigl (\sum A_i\bigr )$ is a a subset of $\Delta ^{m-1} \times \Delta ^{n-1}$. If $A_i = \Delta ^{m-1}$ for all i, then $\mathcal C\bigl (\sum A_i\bigr ) = \Delta ^{m-1} \times \Delta ^{n-1}$.

Example 3.6

If $\Pi ^{m-1}$ is the $(m-1)$-dimensional permutohedron (see Example 3.2), then $\mathcal C(\Pi ^{m-1})$ can be written as

$$\begin{aligned} \mathcal C(\Pi ^{m-1}) = \bigcup _{1 \le i < j \le m} \{ (e_i,f_{ij}), (e_j,f_{ij}) \} \subset \mathbb R^m {\times } \mathbb R^{\left( {\begin{array}{c}m\\ 2\end{array}}\right) }. \end{aligned}$$

We say that two fine mixed subdivisions $\mathscr {T}$ and $\mathscr {T}'$ differ by a flip if the triangulations $\mathcal C(\mathscr {T})$ and $\mathcal C(\mathscr {T}')$ differ by a flip. We define the flip graph analogously.

Remark 3.7

Cayley embeddings of zonotopes are Lawrence polytopes. The connectivity of flip graphs on zonotopal tilings of zonotopes can be equivalently stated as the connectivity of flip graphs on triangulations of Lawrence polytopes.

4 A Zonotope with Disconnected Flip Graph

We are now ready to construct a zonotope and a nontrivial component of its flip graph. Explicitly, this zonotope is the three-dimensional permutohedron, but with each of its generating vectors repeated a large number of times. The idea of the proof will be to generate certain random tilings of this zonotope, and then show that most of the generated tilings will be in different components of the flip graph. We begin by developing the machinery needed to generate the random tilings.

4.1 Zonotopal Tilings of the 3-Permutohedron

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.

We first set up some notation. For a set S, let $\Gamma _S^k$ denote the set of all ordered k-tuples $(i_1,\cdots ,i_k)$ of distinct $i_1$, $\dots ,$ $i_k \in S$ under the equivalence relation $(i_1,\cdots ,i_k) \sim (i_2,\cdots ,i_k,i_1)$. We will use $(i_1 \cdots i_k)$ to denote the equivalence class of $(i_1,\cdots ,i_k)$ in $\Gamma _S^k$. We write—$(i_1 \cdots i_k)$ to denote $(i_k \cdots i_1)$. We abbreviate $\Gamma _{[n]}^k$ as $\Gamma _n^k$.

The main object we will focus on is the Cayley embedding of the 3-dimensional permutohedron. This object was described in Example 3.6; here we will redefine it using slightly different indices. Let $\{e_1,\cdots ,e_4\}$ be the standard basis for $\mathbb R^4$ and let $\{ f_\alpha \}_{\alpha \in \Gamma _4^2}$ be the standard basis for $\mathbb R^6$. Then

$$\begin{aligned} \Pi ^3 \,:= \bigcup _{(ij) \in \Gamma _4^2} \{ (e_i,f_{(ij)}), (e_j,f_{(ij)}) \} \subset \mathbb R^4 \times \mathbb R^6 \end{aligned}$$

is the (Cayley embedding of the) 3-dimensional permutohedron. (Note we have written $\Pi ^3$ instead of $\mathcal C(\Pi ^3)$ here for convenience.)

Our proof will focus on specific circuits of size 6 in the permutohedron. For any $(ijk) \in \Gamma _4^3$, we have a circuit $X_{(ijk)} = (X_{(ijk)}^+,X_{(ijk)}^-)$ in $\Pi ^3$ with affine dependence relation

$$\begin{aligned} \bigl (e_i,f_{(ij)}\bigr ) - \bigl (e_j,f_{(ij)}\bigr ) + \bigl (e_j,f_{(jk)}\bigr ) - \bigl (e_k,f_{(jk)}\bigr ) + \bigl (e_k,f_{(ki)}\bigr ) - \bigl (e_i,f_{(ki)}\bigr ) \end{aligned}$$

and $X_{(ijk)}^+,X_{(ijk)}^-$ defined in terms of this affine relation. Define $\mathscr {T}_{(ijk)} := \mathscr {T}_{X_{(ijk)}}^+$. Note that $X_{-\gamma } = (X^-_{\gamma }, X^+_{\gamma })$ and hence $\mathscr {T}_{-\gamma } = \mathscr {T}_{X_{\gamma }}^-$.

Remark 4.1

In the zonotope setting, each circuit $X_{(ijk)}$ “corresponds” to a pair of opposite hexagonal facets of the 3-permutohedron, namely the facets generated by $\{e_i-e_j, e_j-e_k, e_k-e_i\}$; see Fig. 1. The correspondence is in the following sense: In any tight zonotopal tiling $\mathscr {S}$ of the permutohedron, these facets are tiled in the same way. The Cayley embedding of $\mathscr {S}$ is a triangulation $\mathcal C(\mathscr {S})$ of $\Pi ^3$, and the tiling of the aforementioned facets is determined by the triangulation induced by $\mathcal C(\mathscr {S})$ on the circuit $X_{(ijk)}$.

Fig. 1

Facets (outlined in red) corresponding to the circuit $X_{(ijk)}$

We will now construct eight different tilings of $\Pi ^3$, each indexed by a different element of $\Gamma _4^3$. Fix some $\gamma = (ijk) \in \Gamma _4^3$. Let $\omega :\Pi ^3 \rightarrow \mathbb R$ be the function with

$$\begin{aligned} \omega \bigl (e_i,f_{(ij)}\bigr ) = \omega \bigl (e_j,f_{(jk)}\bigr ) = \omega \bigl (e_k,f_{(ki)}\bigr ) = 1 \end{aligned}$$

and $\omega (x) = 0$ for all other $x \in \Pi ^3$. We define $\mathscr {T}_{\Pi ^3}^\gamma \,:=\, \mathscr {S}_{\Pi ^3}^\omega$. It is easy to check (using Theorem 3.3, for example) that $\mathscr {T}_{\Pi ^3}^\gamma$ is a triangulation.

Let us take a closer look at $\mathscr {T}_{\Pi ^3}^\gamma$ where $\gamma = (ijk)$. For each $\gamma ' \in \Gamma _4^3$, the circuit $X_{\gamma '}$ is a face of $\Pi ^3$, and thus $\mathscr {T}_{\Pi ^3}^\gamma$ contains the triangulation it induces on $X_{\gamma '}$. From the definition of $\omega$, we have

$$\begin{aligned} \omega \bigl (e_i,f_{(ij)}\bigr ) - \omega \bigl (e_j,f_{(ij)}\bigr )&+\, \omega \bigl (e_j,f_{(jk)}\bigr ) - \omega \bigl (e_k,f_{(jk)}\bigr ) \\&+\,\omega \bigl (e_k,f_{(ki)}\bigr ) - \omega \bigl (e_i,f_{(ki)}\bigr ) > 0 \end{aligned}$$

and if l is the element of $[4] \,\backslash \,\{i,j,k\}$, we have

$$\begin{aligned} \omega \bigl (e_i,f_{(ij)}\bigr ) - \omega \bigl (e_j,f_{(ij)}\bigr )&+ \omega \bigl (e_j,f_{(jl)}\bigr ) - \omega \bigl (e_l,f_{(jl)}\bigr )\\&+ \omega \bigl (e_l,f_{(li)}\bigr ) - \omega \bigl (e_i,f_{(li)}\bigr )> 0, \\ \omega \bigl (e_j,f_{(jk)}\bigr ) - \omega \bigl (e_k,f_{(jk)}\bigr )&+ \omega \bigl (e_k,f_{(kl)}\bigr ) - \omega \bigl (e_l,f_{(kl)}\bigr )\\&+ \omega \bigl (e_l,f_{(lj)}\bigr ) - \omega \bigl (e_j,f_{(lj)}\bigr )> 0, \\ \omega \bigl (e_k,f_{(ki)}\bigr ) - \omega \bigl (e_i,f_{(ki)}\bigr )&+ \omega \bigl (e_i,f_{(il)}\bigr ) - \omega \bigl (e_l,f_{(il)}\bigr )\\&+ \omega \bigl (e_l,f_{(lk)}\bigr ) - \omega \bigl (e_k,f_{(lk)}\bigr ) > 0. \end{aligned}$$

Thus, by Proposition 2.4, $\mathscr {T}_{\Pi ^3}^\gamma$ induces the following triangulations on the circuits $X_{\gamma '}$:

$$\begin{aligned} \mathscr {T}_{(ijk)}, \mathscr {T}_{(ijl)}, \mathscr {T}_{(jkl)}, \mathscr {T}_{(kil)} \subseteq \mathscr {T}_{\Pi ^3}^\gamma . \end{aligned}$$

(4.1)

4.2 A Group Action on $\Gamma _4^3$

The key property of $\mathscr {T}_{\Pi ^3}^\gamma$ 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 3-permutohedron with some $\mathscr {T}_{\Pi ^3}^\gamma$ so that in the end, no circuit of size six can be flipped. To help with this construction, we will take some time to define a group action on $\Gamma _4^3$.

For each $\gamma = (ijk) \in \Gamma _4^3$, we define a function $o_\gamma :\left( {\begin{array}{c}[4]\\ 3\end{array}}\right) \rightarrow \Gamma _4^3$ by

$$\begin{aligned} o_\gamma (\{i,j,k\})&= (ijk), \\ o_\gamma (\{i,j,l\})&= (ijl), \\ o_\gamma (\{j,k,l\})&= (jkl), \\ o_\gamma (\{k,i,l\})&= (kil), \end{aligned}$$

where $\{l\} = [4] \,\backslash \,\{i,j,k\}$. The function $o_\gamma$ is a way to “orient” each element of $\left( {\begin{array}{c}[4]\\ 3\end{array}}\right)$ with respect to $\gamma$. Recalling Eq. (4.1), $o_\gamma$ is defined in such a way that $\mathscr {T}_{o_\gamma (S)} \subseteq \mathscr {T}_{\Pi ^3}^\gamma$ for all $S \in \left( {\begin{array}{c}[4]\\ 3\end{array}}\right)$. It is easy to check that $\gamma$ is determined by $o_\gamma$.

Now, we will map each $\alpha \in \Gamma _4^2$ to a permutation $\pi _\alpha :\Gamma _4^3 \rightarrow \Gamma _4^3$. This map is completely determined by the following rules: For any distinct $i,j,k,l \in [4]$, we have

$$\begin{aligned} \pi _{(ij)}(ijk)&= (jil), \\ \pi _{(kl)}(ijk)&= (ijl). \end{aligned}$$

We can check that for all distinct $i,j,k,l \in [4]$ and $\gamma \in \Gamma _4^3$, we have

$$\begin{aligned} \begin{aligned} o_{\pi _{(ij)}\gamma }(\{i,j,k\})&= -o_\gamma (\{i,j,k\}), \\ o_{\pi _{(ij)}\gamma }(\{i,j,l\})&= -o_\gamma (\{i,j,l\}), \\ o_{\pi _{(ij)}\gamma }(\{j,k,l\})&= o_\gamma (\{j,k,l\}), \\ o_{\pi _{(ij)}\gamma }(\{k,i,l\})&= o_\gamma (\{k,i,l\}). \end{aligned} \end{aligned}$$

(4.2)

In other words, $\pi _{(ij)}$ reverses the value of $o_\gamma$ on $S \in \left( {\begin{array}{c}[4]\\ 3\end{array}}\right)$ if $\{i,j\} \subseteq S$ and leaves it the same otherwise. Let $G_{\Gamma _4^3}$ be the permutation group of $\Gamma _4^3$ generated by all the $\pi _\alpha$.

Proposition 4.2

The following are true.

• 1. Every element of $G_{\Gamma _4^3}$ is an involution, and $G_{\Gamma _4^3}$ is abelian and transitive on $\Gamma _4^3$.

• 2. For $l \in [4]$, let $H_l$ be the subgroup of $G_{\Gamma _4^3}$ generated by $\pi _{(il)}$ for all $i \in [4] \,\backslash \,\{l\}$. Let $i,j,k \in [4] \,\backslash \,\{l\}$ be distinct, and let $\Gamma _4^3(ijk)$ be the set of all $\gamma \in \Gamma _4^3$ such that $o_\gamma (\{i,j,k\}) = (ijk)$. Then $\Gamma _4^3(ijk)$ is an orbit of $H_l$.

Proof

Since each $\gamma$ is determined by $o_\gamma$, we can view $G_{\Gamma _4^3}$ as an action on the set of functions $o_\gamma$. It is then clear from (4.2) that we can embed $G_{\Gamma _4^3}$ as a subgroup of $\mathbb Z_2^4$. This implies that every element of $G_{\Gamma _4^3}$ is an involution and $G_{\Gamma _4^3}$ is abelian. It is also easy to check from (4.2) that every element of $\Gamma _4^3$ has orbit of size 8, and hence $G_{\Gamma _4^3}$ is transitive.

From (4.2), we see that $H_l$ maps $\Gamma _4^3(ijk)$ to itself and every element of $\Gamma _4^3(ijk)$ has orbit of size 4 under $H_l$. Since $|\Gamma _4^3(ijk) | = 4$, $\Gamma _4^3(ijk)$ is an orbit of $H_l$. $\square$

4.3 A Zonotope and a Component of Its Flip Graph

We are now ready to go into the main proof. Let N be a positive integer to be determined later. For each distinct $i,j \in [4]$ and $-N \le r \le N$, we create a variable $f_{ij}^r$, and we make the identification of variables

$$\begin{aligned} f_{ij}^r = f_{ji}^{-r}. \end{aligned}$$

We think of the $f_{ij}^r$ as copies of the variable $f_{(ij)}$ defined in Sect. 4.1. Let $\{f_{ij}^r\}_{1 \le i < j \le 4, -N \le r \le N}$ be the standard basis for $\mathbb R^{6(2N+1)}$. Let

$$\begin{aligned} \Pi \,:= \bigcup _{1 \le i < j \le 4} \bigcup _{-N \le r \le N} \{(e_i,f_{ij}^r), (e_j,f_{ij}^r) \} \subset \mathbb R^4 \times \mathbb R^{6(2N+1)} \end{aligned}$$

(4.3)

be the 3-dimensional permutohedron with $2N+1$ copies of each generating vector. As a zonotope, $\Pi$ is the unit 3-permutohedron scaled by $2N+1$. We now prove our first main result.

Theorem 4.3

For large enough N, the flip graph of $\Pi$ is not connected.

We will use the following lemma which identifies a component of the flip graph of $\Pi$ based on the triangulations of certain circuits of $\Pi$. For distinct $i,j,k \in [4]$ and for any $-N \le r, s, t \le N$, let $X_{ijk}^{rst} = ((X_{ijk}^{rst})^+, (X_{ijk}^{rst})^-)$ be the circuit with affine dependence relation

$$\begin{aligned} (e_i,f_{ij}^r) - (e_j,f_{ij}^r) + (e_j,f_{jk}^s) - (e_k,f_{jk}^s) + (e_k,f_{ki}^t) - (e_i,f_{ki}^t) \end{aligned}$$

and $(X_{ijk}^{rst})^+,(X_{ijk}^{rst})^-$ defined in terms of this relation. Let $\mathscr {T}_{ijk}^{rst} \,:=\, \mathscr {T}_{X_{ijk}^{rst}}^+$.

Lemma 4.4

Let $\mathscr {C}$ be a collection of triangulations of the form $\mathscr {T}_{ijk}^{rst}$ such that for all $\mathscr {T}_{ijk}^{rst} \in \mathscr {C}$ and $\{l\} = [4] \,\backslash \,\{i,j,k\}$, there exist $1 \le u,v,w \le N$ such that

$$\begin{aligned} \mathscr {T}_{ijl}^{rv(-u)}, \mathscr {T}_{jkl}^{sw(-v)}, \mathscr {T}_{kil}^{tu(-w)} \in \mathscr {C}. \end{aligned}$$

Let $\mathcal S_{\mathscr {C}}$ be the set of all triangulations of $\Pi$ which contain every element of $\mathscr {C}$ as a subset. Then $\mathcal S_{\mathscr {C}}$ is closed under flips.

The proof of this lemma follows immediately from the following two facts.

Proposition 4.5

Let $\mathscr {T}$ be a triangulation of $\Pi$ such that $\mathscr {T}_{ijk}^{rst} \subseteq \mathscr {T}$. Let $\mathscr {T}'$ be the result of a flip on $\mathscr {T}$ which is not supported on $X_{ijk}^{rst}$. Then $\mathscr {T}_{ijk}^{rst} \subseteq \mathscr {T}'$.

Proof

Suppose the flip from $\mathscr {T}$ to $\mathscr {T}'$ is supported on $X = (X^+,X^-)$. By Proposition 2.2, if $\sigma \in \mathscr {T}$ and $\sigma \notin \mathscr {T}'$, then $\sigma \supseteq X^-$. Thus, if $\mathscr {T}_{ijk}^{rst} \not \subseteq \mathscr {T}'$, then we must have $X^- \subseteq X_{ijk}^{rst}$.

On the other hand, the only circuits of $\Pi$ are of the form $X_{i'j'k'}^{r's't'}$ or

$$\begin{aligned} \bigl ( \{ (e_{i'}, f_{i'j'}^{r'}) , (e_{j'}, f_{i'j'}^{s'}) \}, \{ (e_{j'}, f_{i'j'}^{r'}) , (e_{i'}, f_{i'j'}^{s'}) \} \bigr ) \end{aligned}$$

for some $i',j',k',r',s',t'$. Of these circuits, the only circuit whose negative part is contained in $X_{ijk}^{rst}$ is $X_{ijk}^{rst}$ itself. Since $X \ne X_{ijk}^{rst}$ by assumption, we must therefore have $X^- \not \subseteq X_{ijk}^{rst}$, and hence $\mathscr {T}_{ijk}^{rst} \subseteq \mathscr {T}'$. $\square$

Proposition 4.6

Let $\mathscr {T}$ be a triangulation of $\Pi$, and suppose that there are distinct $i,j,k,l \in [4]$ and $1 \le r,s,t,u,v,w \le N$ such that

$$\begin{aligned} \mathscr {T}_{ijl}^{rv(-u)}, \mathscr {T}_{jkl}^{sw(-v)}, \mathscr {T}_{kil}^{tu(-w)} \subseteq \mathscr {T}. \end{aligned}$$

Then $\mathscr {T}$ does not have a flip supported on $X_{ijk}^{rst}$.

Proof

Consider the cell

$$\begin{aligned} C :\,= \bigl \{ (e_i, f_{il}^u), (e_l, f_{il}^u), (e_j, f_{jl}^v), (e_l, f_{jl}^v), (e_k, f_{kl}^w), (e_l, f_{kl}^w) \bigr \}. \end{aligned}$$

C is a face of $\Pi$, as seen by the linear functional $\phi$ such that

$$\begin{aligned} \phi (e_a, 0)&= \ \; \; 0 \quad \text { for all } a, \\ \phi (0,f_{ab}^p)&= {\left\{ \begin{array}{ll} -1 &{} \hbox { if } \, f_{ab}^p = f_{il}^u, f_{jl}^v, \, \hbox { or } \, f_{kl}^w, \\ 0 &{} \hbox { otherwise }. \end{array}\right. } \end{aligned}$$

Since C is a simplex, we must have $C \in \mathscr {T}$. Thus, there is some maximal simplex $\tau \in \mathscr {T}$ such that $C \subseteq \tau$. Since $\tau$ is maximal, it must contain one element from each of the sets

$$\begin{aligned} \{ (e_i,f_{ij}^r), (e_j,f_{ij}^r) \}, \{ (e_j,f_{jk}^s), (e_k,f_{jk}^s) \}, \{ (e_k,f_{ki}^t), (e_i,f_{ki}^t)\}. \end{aligned}$$

Suppose $(e_i,f_{ij}^r) \in \tau$. Then since $C \subseteq \tau$, we have

$$\begin{aligned} (X_{ijl}^{rv(-u)})^+ = \{ (e_i,f_{ij}^r), (e_j, f_{jl}^v), (e_l, f_{li}^{-u}) \} \subseteq \tau . \end{aligned}$$

On the other hand, since $\mathscr {T}_{ijl}^{rv(-u)} \subseteq \mathscr {T}$ by assumption, we have $(X_{ijl}^{rv(-u)})^- \in \mathscr {T}$. This is a contradiction, because the opposite parts of a circuit cannot both be cells of a triangulation (since the interiors of these cells intersect). Hence $(e_j,f_{ij}^r)\in \tau$. Similarly, $(e_k,f_{jk}^s) \in \tau$ and $(e_i,f_{ki}^t) \in \tau$. Hence, $(X_{ijk}^{rst})^- \subseteq \tau$.

We thus have $(X_{ijk}^{rst})^- \subseteq \tau$ and $| X_{ijk}^{rst} \cap \tau | = 3 < | X_{ijk}^{rst} | - 2$. By Proposition 2.3, $\mathscr {T}$ does not have a flip supported on $X_{ijk}^{rst}$. $\square$

Theorem 4.3 follows from Lemma 4.4 if there is some $\mathscr {C}$ for which $\mathcal S_{\mathscr {C}}$ is neither empty nor the whole set of triangulations of $\Pi$. We show this in the next section.

4.4 Construction of $\mathscr {C}$ and Some $\mathscr {T} \in \mathcal S_{\mathscr {C}}$

We first consider the regular subdivision $\mathscr {S}_\Pi ^\omega$ where $\omega :\Pi \rightarrow \mathbb R$ is a function such that

$$\begin{aligned} \omega (e_i, f_{ij}^r) - \omega (e_j, f_{ij}^r) = r \end{aligned}$$

for all distinct $i,j \in [4]$ and $-N \le r \le N$. The cells of $\mathscr {S}_\Pi ^\omega$ are described as follows.

Proposition 4.7

Let $\mathscr {X}$ be the set of $x = (x_1,x_2,x_3,x_4) \in \mathbb R^4$ such that $x_1 + \cdots + x_4 = 0$ and if ijkl is a permutation of [4] such that $x_i \ge x_j \ge x_k \ge x_l$, then $x_i - x_j$, $x_j - x_k$, and $x_k - x_l$ are integers at most N. Let $\mathscr {X}^*$ be the set of $x \in \mathscr {X}$ such that $|x_i-x_j | \le N$ for all $i,j \in [n]$. The following are true.

• 1. The map $C(x) = \{ (e_i, f_{ij}^r) : x_i - x_j \ge r \}$ is a bijection from $\mathscr {X}$ to the maximal cells of $\mathscr {S}_\Pi ^\omega$.

• 2. If $x \in \mathscr {X}^*$, then C(x) is the Cayley embedding of a translated 3-permutohedron. Specifically, $C(x) = \Pi (x) \cup D$, where

  $$\begin{aligned} \Pi (x) \,:= \bigcup _{1 \le i < j \le 4} \bigl \{\bigl (e_i, f_{ij}^{x_i - x_j}\bigr ), \bigl (e_j, f_{ij}^{x_i-x_j}\bigr ) \bigr \} \end{aligned}$$

  and D is a simplex affinely independent to $\Pi (x)$.

• 3. Let $x \in \mathscr {X} \,\backslash \,\mathscr {X}^*$. Suppose $F_1$, $\dots ,$ $F_k$ are faces of C(x) and $\mathscr {T}_1$, $\dots ,$ $\mathscr {T}_k$ are triangulations of these faces, respectively, which agree on intersections of these faces. Then there is a triangulation of C(x) which contains $\mathscr {T}_1$, $\dots ,$ $\mathscr {T}_k$ as subsets.

Proof

We first prove Part 1. By Theorem 3.3, a set $C \subseteq \Pi$ is a cell of $\mathscr {S}_\Pi ^\omega$ if and only if there is some $x \in \mathbb R^4$ such that

$$\begin{aligned} C&= \bigl \{ (e_i, f_{ij}^r) : x_i - \omega (e_i,f_{ij}^r) \ge x_j - \omega (e_j,f_{ij}^r) \bigr \} \\&= \bigl \{ (e_i, f_{ij}^r) : x_i - x_j \ge r \bigr \} = C(x). \end{aligned}$$

Moreover, if C is maximal, then there are pairs $(i,j),(i',j'),(i'',j'')$ such that $\{e_i-e_j, e_{i'}-e_{j'}, e_{i''}-e_{j''}\}$ is linearly independent and

$$\begin{aligned} \bigl \{ (e_i,f_{ij}^r), (e_j,f_{ij}^r), (e_{i'},f_{i'j'}^{r'}), (e_{j'},f_{i'j'}^{r'}), (e_{i''},f_{i''j''}^{r''}), (e_{j''},f_{i''j''}^{r''}) \bigr \} \subseteq C \end{aligned}$$

(4.4)

for some $-N \le r,r',r'' \le N$. If $x \in \mathbb R^4$ is such that $C = C(x)$ for this maximal cell C, then (4.4) implies $x_i-x_j = r$, $x_{i'}-x_{j'} = r'$, and $x_{i''}-x_{j''}=r''$. By the linear independence of $\{e_i-e_j, e_{i'}-e_{j'}, e_{i''}-e_{j''}\}$, there is a unique x for which these equalities hold and $x_1 + \cdots + x_4 = 0$. Moreover, this x is contained in $\mathscr {X}$. Finally, for any $x \in \mathscr {X}$, C(x) is a maximal cell of $\mathscr {S}_\Pi ^\omega$. This proves Part 1. Part 2 follows immediately.

The only nontrivial case of Part 3 is when x satisfies $x_i \ge x_j \ge x_k \ge x_l$ for some permutation ijkl of [4], $x_i - x_k \le N$, $x_j - x_l \le N$, and $x_i - x_l > N$. (A zonotopal tile corresponding to such a cell is highlighted in yellow in Fig. 2.) In this case C(x) is of the form $X_{ijk}^{rst} \cup X_{jkl}^{suv} \cup D$, where D is a simplex affinely independent to $X_{ijk}^{rst} \cup X_{jkl}^{suv}$. By Proposition 5.9 (or by an easy check), any triangulations of $X_{ijk}^{rst}$ and $X_{jkl}^{suv}$ can be extended to a triangulation of $X_{ijk}^{rst} \cup X_{jkl}^{suv}$, and hence to a triangulation of C(x). $\square$

Remark 4.8

In terms of mixed subdivisions, the coherent mixed subdivision associated to $\omega$ is given by tiling the large permutohedron $\Pi$ with smaller unit permutohedra (the cells corresponding to $\mathscr {X}^*$) and pieces of permutohedra (the tiles corresponding to $\mathscr {X} \,\backslash \,\mathscr {X}^*$). Each point $x \in \mathscr {X}$ is the center of the zonotopal tile corresponding to C(x).

Fig. 2

A portion of the tiling associated to $\omega$

We now give a brief overview of the rest of the proof. The goal is to construct a triangulation which refines $\mathscr {S}_{\Pi }^\omega$ and which is sufficiently “complicated”. The idea is to start with a known triangulation and apply the group action $G_{\Gamma _4^3}$ to the triangulations of the cells C(x) in a way that produces another triangulation of $\Pi$. As we will show, by randomly doing this process we can produce triangulations from our original triangulation that cannot be produced through sequences of flips.

For each $x \in \mathscr {X}^*$, we have an affine isomorphism $\Pi ^3 \rightarrow \Pi (x)$ given by $f_{(ij)} \mapsto f_{ij}^{x_i-x_j}$. For each $\gamma \in \Gamma _4^3$, let $\mathscr {T}_{\Pi (x)}^\gamma$ be the image of $\mathscr {T}_{\Pi ^3}^\gamma$ under this isomorphism.

We will now choose a random triangulation of every C(x), $x \in \mathscr {X}^*$, as follows:

• 1. For each $1 \le i < j \le 4$ and $-N \le r \le N$, let $g_{ij}^r = g_{ji}^{-r}$ be an independent random element of $G_{\Gamma _4^3}$ which is 1 with probability 1 / 2 and $\pi _{(ij)}$ with probability 1 / 2.

• 2. For each $x \in \mathscr {X}^*$, triangulate $\Pi (x)$ by $\mathscr {T}_{\Pi (x)}^{\gamma (x)}$, where

  $$\begin{aligned} \gamma (x) \,:=\, \Big ( \prod _{1 \le i < j \le 4} g_{ij}^{x_i-x_j} \Big ). \end{aligned}$$

  (123)

• 3. Extend $\mathscr {T}_{\Pi (x)}^{\gamma (x)}$ uniquely to a triangulation $\mathscr {T}_{C(x)}$ of C(x).

Proposition 4.9

For any two $x,x' \in \mathscr {X}^*$, the triangulations $\mathscr {T}_{C(x)}$ and $\mathscr {T}_{C(x')}$ agree on the common face of C(x) and $C(x')$.

Proof

The only non-trivial case is when $C(x) \cap C(x')$ contains a circuit $X_{ijk}^{rst}$. We need to check that $\mathscr {T}_{C(x)}$ and $\mathscr {T}_{C(x')}$ agree on this circuit. If $X_{ijk}^{rst} \subseteq C(x) \cap C(x')$, then

$$\begin{aligned} x_i - x_j = x'_i - x'_j&= r, \\ x_j - x_k = x'_j - x'_k&= s, \\ x_k - x_i = x'_k = x'_i&= t. \end{aligned}$$

On the other hand, by Proposition 4.2 (2), $o_{\gamma (x)}(\{i,j,k\})$ depends only on $g_{ij}^{x_i-x_j},g_{jk}^{x_j-x_k}$, and $g_{ki}^{x_k-x_i}$. It follows that $o_{\gamma (x)}(\{i,j,k\}) = o_{\gamma (x')}(\{i,j,k\})$. Thus $\mathscr {T}_{\Pi (x)}^{\gamma (x)}$ and $\mathscr {T}_{\Pi (x')}^{\gamma (x')}$ contain the same triangulation of $X_{ijk}^{rst}$, as desired. $\square$

By Proposition 4.9 and Proposition 4.7 (3), we can thus extend the above triangulations of the C(x) to a full triangulation of $\Pi$. Call this triangulation $\mathscr {T}$.

Let $\mathscr {C}$ be the collection of all triangulations $\mathscr {T}_{ijk}^{rst} \subseteq \mathscr {T}$ with $i,j,k \in [4]$ distinct, $-N \le r,s,t \le N$, and $r+s+t=0$. We prove that $\mathscr {C}$ satisfies the hypotheses of Lemma 4.4. We will actually prove the following stronger statement, which we will need in the next section. (For the current proof, we only need the second sentence of (B).)

Proposition 4.10

For large enough N, with probability greater than 0, $\mathscr {T}$ and $\mathscr {C}$ satisfy the following:

• (A) For every distinct $i,j,k \in [4]$ and $-N \le r \le N$, there exist $-N \le s,t \le N$ such that $\mathscr {T}_{ijk}^{rst} \in \mathscr {C}$.

• (B) For every $\mathscr {T}_{ijk}^{rst} \in \mathscr {C}$ and $\gamma \in \Gamma _4^3(ijk)$, there exists $x \in \mathscr {X}^*$ such that $X_{ijk}^{rst} \subseteq \Pi (x)$ and $\mathscr {T}[\Pi (x)] = \mathscr {T}_{\Pi (x)}^\gamma$. In particular, when $\gamma = (ijk)$, this implies there is some $-N \le u,v,w \le N$ such that

  $$\begin{aligned} \mathscr {T}_{ijl}^{rv(-u)}, \mathscr {T}_{jkl}^{sw(-v)}, \mathscr {T}_{kil}^{tu(-w)} \in \mathscr {C}, \end{aligned}$$

  where $\{l\} = [4] \,\backslash \,\{i,j,k\}$.

Proof

First, note that for any distinct $i,j,k \in [4]$ and $-N \le r,s,t \le N$ with $r+s+t=0$, there is some $x \in \mathscr {X}^*$ such that $X_{ijk}^{rst} \subseteq \Pi (x)$. Hence $\mathscr {C}$ contains a triangulation of $X_{ijk}^{rst}$. Letting $\gamma = g_{ij}^r g_{jk}^s g_{ki}^t (123)$, we have

$$\begin{aligned} \mathscr {T}_{ijk}^{rst} \in \mathscr {C} \quad \text {if and only if } o_\gamma (\{i,j,k\}) = (ijk). \end{aligned}$$

(4.5)

We first bound the probability that (A) does not hold. Fix distinct $i,j,k \in [4]$ and $-N \le r \le N$. Let H be the set of all ordered pairs $(s,t) \in [-N,N]^2$ with $r+s+t=0$. Note that $|H | \ge N$. For each $(s,t) \in H$ and $\gamma (s,t) \,:=\, g_{ij}^r g_{jk}^s g_{ki}^t (123)$, it is easy to see from the proof of Proposition 4.2 that $o_{\gamma (s,t)}(\{i,j,k\}) = (ijk)$ with probability 1 / 2. In fact, this probability does not change if we fix $g_{ij}^r$, so for all $(s,t) \in H$ these probabilities are mutually independent. Now, from (4.5), there does not exist $(s,t) \in H$ such that $\mathscr {T}_{ijk}^{rst} \in \mathscr {C}$ if and only if $o_{\gamma (s,t)}(\{i,j,k\}) \ne (ijk)$ for all $(s,t) \in H$. The probability this happens is

$$\begin{aligned} \Big ( \frac{1}{2} \Big )^{|H |} \le \Big ( \frac{1}{2} \Big )^{N}. \end{aligned}$$

By the union bound, the probability that this happens for some distinct $i,j,k \in [4]$ and $-N \le r \le N$ is at most

$$\begin{aligned} 24(2N+1)\Big ( \frac{1}{2} \Big )^{N}. \end{aligned}$$

(4.6)

This gives an upper bound on the probability of (A) not happening.

We now do the same for (B). Fix $\mathscr {T}_{ijk}^{rst} \in \mathscr {C}$ and $\gamma \in \Gamma _4^3(ijk)$. Let H be the set of all $x \in \mathscr {X}^*$ such that $X_{ijk}^{rst} \subseteq \Pi (x)$. Note that $|H | \ge N$. Let $\{l\} = [4] \,\backslash \,\{i,j,k\}$.

We have $\mathscr {T}_{ijk}^{rst} \in \mathscr {C}$, which happens if and only if $g_{ij}^r g_{jk}^s g_{ki}^t (123) \in \Gamma _4^3(ijk)$. Suppose we fix $g_{ij}^r,g_{jk}^s$, and $g_{ki}^t$ such that $g_{ij}^r g_{jk}^s g_{ki}^t (123) \in \Gamma _4^3(ijk)$. Then for each $x \in H$, it follows from Proposition 4.2 and the definition of $\gamma (x)$ that $\gamma (x) = \gamma$ with probability 1 / 4. Moreover, since $(x_i - x_l, x_j - x_l, x_k - x_l)$ is different for each $x \in H$, these probabilities are mutually independent for all $x \in H$. Now, for each $x \in \mathscr {X}^*$ we have $\mathscr {T}[\Pi (x)] = \mathscr {T}_{\Pi (x)}^{\gamma (x)}$. Thus the probability that there is no $x \in H$ with $\mathscr {T}[\Pi (x)] = \mathscr {T}_{\Pi (x)}^\gamma$ is

$$\begin{aligned} \Big ( \frac{3}{4} \Big )^{|H |} \le \Big ( \frac{3}{4} \Big )^{N}. \end{aligned}$$

The probability that this occurs for some distinct $i,j,k \in [4]$, $-N \le r,s,t \le N$ with $r+s+t=0$, and $\gamma \in \Gamma _4^3(ijk)$ is thus at most

$$\begin{aligned} 96(2N+1)^2\Big ( \frac{3}{4} \Big )^{N}. \end{aligned}$$

(4.7)

Hence, the probability that either (A) or (B) does not hold is at most

$$\begin{aligned} 24(2N+1)\Big ( \frac{1}{2} \Big )^{N} + 96(2N+1)^2\Big ( \frac{3}{4} \Big )^{N}, \end{aligned}$$

which for large enough N (specifically, $N \ge 48$) is less than 1. $\square$

Thus there exists $\mathscr {C}$ which satisfies the hypotheses of Lemma 4.4 and $\mathscr {T} \in \mathcal S_{\mathscr {C}}$. There are triangulations of $\Pi$ which are not in $\mathcal S_{\mathscr {C}}$; for example, a different choice of the $g_{ij}^r$ would yield a triangulation which does not contain every element of $\mathscr {C}$. This proves Theorem 4.3.

5 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 $\mathscr {T}$ in different directions in this space.

5.1 A Component of the Flip Graph

For each $\alpha \in \Gamma _5^2$, construct a finite set $\Delta _\alpha$ of variables. We will determine the size of this set later. Let $\Delta ^4 \,:=\, \{e_1,\dots ,e_5\}$ be the standard basis for $\mathbb R^5$, and let $\Delta ^{n-1} {:=} \bigcup _{\alpha \in \Gamma _5^2} \Delta _\alpha$ be the standard basis for $\mathbb R^n$, where $n = \sum _{\alpha \in \Gamma _5^2} |\Delta _\alpha |$. Let $A \,:=\, \Delta ^4 \times \Delta ^{n-1}$. We prove the following.

Theorem 5.1

For large enough n, the flip graph of A is not connected.

We first identify a component of the flip graph of A. For distinct $i_1$, ..., $i_t \in [5]$ and distinct $f_1$, ..., $f_t \in \Delta ^{n-1}$, let

$$\begin{aligned} X_{i_1 \cdots i_t}^{f_1 \cdots f_t} = \bigl ((X_{i_1 \cdots i_t}^{f_1 \cdots f_t})^+,(X_{i_1 \cdots i_t}^{f_1 \cdots f_t})^-\bigr ) \end{aligned}$$

be the circuit in A with affine dependence relation

$$\begin{aligned} (e_{i_1}, f_1) - (e_{i_2}, f_1) + (e_{i_2}, f_2) - (e_{i_3}, f_3) + \cdots + (e_{i_t}, f_t) - (e_{i_1}, f_t) \end{aligned}$$

and $(X_{i_1 \cdots i_t}^{f_1 \cdots f_t})^+,(X_{i_1 \cdots i_t}^{f_1 \cdots f_t})^-$ defined in terms of this relation. Let

$$\begin{aligned} \mathscr {T}_{i_1 \cdots i_t}^{f_1 \cdots f_t} \,:=\, \mathscr {T}_{X_{i_1 \cdots i_t}^{f_1 \cdots f_t}}^+. \end{aligned}$$

If $f_1 \in \Delta _{(i_1i_2)}$, $f_2 \in \Delta _{(i_2i_3)}$, ..., $f_t \in \Delta _{(i_ti_1)}$, then we call the circuit $X_{i_1 \cdots i_t}^{f_1 \cdots f_t}$ and the triangulation $\mathscr {T}_{i_1 \cdots i_t}^{f_1 \cdots f_t}$ zonotopal. We now identify a component of the flip graph of A.

Lemma 5.2

For each $S \in \left( {\begin{array}{c}[5]\\ 4\end{array}}\right)$ and distinct i, $j \in S$, let $\mathscr {C}_{S,i,j}$ be a collection of zonotopal triangulations of the form $\mathscr {T}_{i_1i_2i_3}^{f_1f_2f_3}$ where $i_1,i_2,i_3 \in S$. Assume that for each $S \in \left( {\begin{array}{c}[5]\\ 4\end{array}}\right)$ and distinct $i,j \in S$, and for each $\mathscr {T}_{i_1i_2i_3}^{f_1 f_2 f_3} \in \mathscr {C}_{S,i,j}$, the following are true.

• 1. If $\{i_4\} = S \,\backslash \,\{i_1,i_2,i_3\}$, then there exist $f_1' \in \Delta _{(i_1i_4)}$, $f_2' \in \Delta _{(i_2i_4)}$, and $f_3' \in \Delta _{(i_3i_4)}$ such that

  $$\begin{aligned} \mathscr {T}_{i_1i_2i_4}^{f_1f_2'f_1'}, \mathscr {T}_{i_2i_3i_4}^{f_2f_3'f_2'}, \mathscr {T}_{i_3i_1i_4}^{f_3f_1'f_3'} \in \mathscr {C}_{S,i,j}. \end{aligned}$$

• 2. If $i_1 = i$, $i_2 = j$, and $i_4$ is as above, then there exist $f_1' \in \Delta _{(i_1i_4)}$, $f_2' \in \Delta _{(i_2i_4)}$, and $f_3' \in \Delta _{(i_3i_4)}$ such that

  $$\begin{aligned} \mathscr {T}_{i_1i_2i_4}^{f_1f_2'f_1'}, \mathscr {T}_{i_3i_2i_4}^{f_2f_2'f_3'}, \mathscr {T}_{i_1i_3i_4}^{f_3f_3'f_1'} \in \mathscr {C}_{S,i,j}. \end{aligned}$$

In addition, assume the following about the $\mathscr {C}_{S,i,j}$.

• 3. For each distinct $i,j,l \in [5]$ and $f_1 \in \Delta _{(ij)}$, there exist $k \in [5] \,\backslash \,\{i,j,l\}$, $f_2 \in \Delta _{(jk)}$, and $f_3 \in \Delta _{(ki)}$ such that

  $$\begin{aligned} \mathscr {T}_{ijk}^{f_1f_2f_3} \in \mathscr {C}_{[5] \,\backslash \,\{l\}, i, j}. \end{aligned}$$

Let $\mathcal S_{\mathscr {C}}$ be the set of all triangulations $\mathscr {T}$ of A satisfying the following.

• (i) For each $S \in \left( {\begin{array}{c}[5]\\ 4\end{array}}\right)$, distinct $i,j \in S$, and $\mathscr {T}_{i_1i_2i_3}^{f_1 f_2 f_3} \in \mathscr {C}_{S,i,j}$, we have

  $$\begin{aligned} \mathscr {T}_{i_1i_2i_3}^{f_1 f_2 f_3} \subseteq \mathscr {T}. \end{aligned}$$

• (ii) For each $S \in \left( {\begin{array}{c}[5]\\ 4\end{array}}\right)$, distinct $i,j \in S$, and $\mathscr {T}_{ijk}^{f_1 f_2 f_3} \in \mathscr {C}_{S,i,j}$ where $k \in S \,\backslash \,\{i,j\}$, if $\{l\} = [5] \,\backslash \,S$, then for any $f \in \Delta _{(il)}$ we have

  $$\begin{aligned} X_{ijk}^{f_1 f_2 f_3} \,\backslash \,\{(e_i,f_1)\} \cup \{(e_i,f)\} \in \mathscr {T}. \end{aligned}$$

Then $\mathcal S_{\mathscr {C}}$ is closed under flips.

Proof

We need the following two facts about triangulations of A. They are analogous to Propositions 4.5 and 4.6. We defer their proofs to the Appendix.

Proposition 5.3

Let $\mathscr {T}$ be a triangulation of A and let $\mathscr {T}'$ be the result of a flip on $\mathscr {T}$ supported on $X = (X^+,X^-)$. Suppose that $\sigma \in \mathscr {T}$ and $\sigma \notin \mathscr {T}'$, and $G(\sigma )$ is connected. Then $\sigma$ contains a maximal simplex of $\mathscr {T}_X^+$.

Proposition 5.4

Let $\mathscr {T}$ be a triangulation of A. Let $i,j,k,l \in [5]$ be distinct, and for each $\alpha \in \Gamma _{\{i,j,k,l\}}^2$, let $f_\alpha \in \Delta ^{n-1}$. Suppose that

$$\begin{aligned} \mathscr {T}_{jkl}^{f_{(jk)}f_{(kl)}f_{(lj)}}, \mathscr {T}_{kil}^{f_{(ki)}f_{(il)}f_{(lk)}} \subseteq \mathscr {T} \end{aligned}$$

and

$$\begin{aligned} X_{ijl}^{f_{(ij)}f_{(jl)}f_{(li)}} \,\backslash \,\{ (e_i, f_{(ij)}) \} \in \mathscr {T}. \end{aligned}$$

Then $\mathscr {T}$ does not have a flip supported on $X_{ijk}^{f_{(ij)}f_{(jk)}f_{(ki)}}$.

We also need the following two facts about flips on elements of $\mathcal S_{\mathscr {C}}$.

Proposition 5.5

Let $i,j,l \in [5]$ be distinct and let $f_1 \in \Delta _{(ij)}$, $f_2 \in \Delta _{(il)}$. Let $\mathscr {T} \in \mathcal S_\mathscr {C}$. Then $\mathscr {T}$ does not have a flip supported on $X_{ij}^{f_1f_2}$.

Proof

By Property 3 of $\mathscr {C}_{S,i,j}$, there exist $k \in [5] \,\backslash \,\{i,j,l\}$, $f_2' \in \Delta _{(jk)}$, and $f_3' \in \Delta _{(ki)}$ such that $\mathscr {T}_{ijk}^{f_1f_2'f_3'} \in \mathscr {C}_{[5] \,\backslash \,\{l\},i,j}$. By property (ii) of $\mathcal S_{\mathscr {C}}$, it follows that

$$\begin{aligned} \sigma \,:=\, X_{ijk}^{f_1f_2'f_3'} \,\backslash \,\{(e_i,f_1)\} \cup \{(e_i,f_2)\} \in \mathscr {T}. \end{aligned}$$

If $\tau$ is a maximal simplex in $\mathscr {T}$ containing $\sigma$, then $(X_{ij}^{f_1f_2})^- \subseteq \tau$ but $|X_{ij}^{f_1f_2} \cap \tau | = 2$. So by Proposition 2.3, $\mathscr {T}$ does not have a flip supported on $X_{ij}^{f_1f_2}$. $\square$

Proposition 5.6

Let $\mathscr {T}_{i_1i_2i_3}^{f_1 f_2 f_3} \in \mathscr {C}_{S,i,j}$. Let $\mathscr {T} \in \mathcal S_\mathscr {C}$. Then $\mathscr {T}$ does not have a flip supported on $X_{i_1i_2i_3}^{f_1 f_2 f_3}$.

Proof

This is a direct corollary of Property 1 of $\mathscr {C}_{S,i,j}$ and Proposition 5.4. $\square$

We now proceed with the proof. Suppose that $\mathscr {T} \in \mathcal S_\mathscr {C}$, and let $\mathscr {T}'$ be the result of a flip on $\mathscr {T}$ supported on $X = (X^+,X^-)$. We prove that properties (i) and (ii) hold for $\mathscr {T}'$.

Property (i). Suppose that $\mathscr {T}_{i_1i_2i_3}^{f_1 f_2 f_3} \in \mathscr {C}_{S,i,j}$ and $\mathscr {T}_{i_1i_2i_3}^{f_1 f_2 f_3} \not \subseteq \mathscr {T}'$. Without loss of generality, assume that $X_{i_1i_2i_3}^{f_1 f_2 f_3} \,\backslash \,\{(e_{i_1},f_1)\} \notin \mathscr {T}'$. By Proposition 5.3, $X_{i_1i_2i_3}^{f_1 f_2 f_3} \,\backslash \,\{(e_{i_1},f_1)\}$ contains a maximal simplex of $\mathscr {T}_X^+$. This leaves two cases for X.

• Case 1: X has size 4. Thus we can write $X = X_{i_1'i_2'}^{f_1'f_2'}$ for some $i_1',i_2' \in \{i_1,i_2,i_3\}$, $f_1' \in \Delta _{(i_1'i_2')}$, and $f_2' \in \Delta _{(i_1'i_3')}$, where $\{i_3'\} = \{i_1,i_2,i_3\} \,\backslash \,\{i_1',i_2'\}$. However, this contradicts Proposition 5.5. So we cannot have $|X | = 4$.

• Case 2: $X = X_{i_1i_2i_3}^{f_1 f_2 f_3}$. This contradicts Proposition 5.6.

Hence we must have $\mathscr {T}_{i_1i_2i_3}^{f_1 f_2 f_3} \subseteq \mathscr {T}'$, as desired.

Property (ii). Suppose that $\sigma \,:=\, X_{ijk}^{f_1 f_2 f_3} \,\backslash \,\{(e_i,f_1)\} \cup \{(e_i,f)\} \notin \mathscr {T}'$, with variables as defined in (ii). By Proposition 5.3, $\sigma$ contains a maximal simplex of X. By the same argument as in part (i), we cannot have $|X | = 4$ or $X = X_{ijk}^{f_1 f_2 f_3}$. This leaves

$$\begin{aligned} X = X_{jik}^{ff_3f_2} \end{aligned}$$

as the only possibility.

We show that $\mathscr {T}$ cannot have a flip on this circuit. Let $l' = S \,\backslash \,\{i,j,k\}$. By Property 2 of $\mathscr {C}_{S,i,j}$, there exist $f_1' \in \Delta _{(il')}$, $f_2' \in \Delta _{(jl')}$, $f_3' \in \Delta _{(kl')}$ such that

$$\begin{aligned} \mathscr {T}_{ijl'}^{f_1f_2'f_1'}, \mathscr {T}_{kjl'}^{f_2f_2'f_3'}, \mathscr {T}_{ikl'}^{f_3f_3'f_1'} \in \mathscr {C}_{S,i,j}. \end{aligned}$$

(5.1)

By Property (ii), the first of these inclusions implies that

$$\begin{aligned} X_{ijl'}^{f_1f_2'f_1'} \,\backslash \,\{(e_i,f_1)\} \cup \{(e_i,f)\} \in \mathscr {T} \end{aligned}$$

and hence

$$\begin{aligned} X_{jil'}^{ff_1'f_2'} \,\backslash \,\{(e_j,f)\} \in \mathscr {T}. \end{aligned}$$

This inclusion along with the last two inclusions of (5.1) imply, by Proposition 5.4, that $\mathscr {T}$ does not have a flip on $X_{jik}^{ff_3f_2}$, as desired. $\square$

5.2 Reduction to Zonotopes

Let

$$\begin{aligned} \Pi \,:= \bigcup _{(ij) \in \Gamma _5^2} \bigcup _{f \in \Delta _{(ij)}} \{ (e_i, f), (e_j,f) \} \end{aligned}$$

be the large 4-permutohedron embedded in A. Suppose we have collections $\mathscr {C}_{S,i,j}$ which satisfy the conditions of Lemma 5.2. Let $\mathscr {T}$ be a triangulation of $\Pi$. Notice that if Properties (i) and (ii) of Lemma 5.2 hold for $\mathscr {T}$, then they hold for any collection containing $\mathscr {T}$. In particular, if $\mathscr {T}$ can be extended to a triangulation $\mathscr {T}'$ of A, then we will have some $\mathscr {T}' \in \mathcal S_{\mathscr {C}}$. The next proposition guarantees we can always do this.

Proposition 5.7

If $\mathscr {T}$ is a triangulation of $\Pi$, then there is a triangulation $\mathscr {T}'$ of A with $\mathscr {T} \subseteq \mathscr {T}'$.

Proof

Let $\mathscr {S}_A \,:=\, \mathscr {S}_A^\omega$ be the regular subdivision of A with $\omega :A \rightarrow \mathbb R$ defined as follows. For each distinct $i,j,k \in [5]$ and $f \in \Delta _{(ij)}$, let $\epsilon _{f,k} > 0$ be a generic positive real number, and define

$$\begin{aligned} \omega (e_i,f) = 0, \qquad \omega (e_j,f) = 0, \qquad \omega (e_k,f) = \epsilon _{f,k}. \end{aligned}$$

Then $\mathscr {S}_A$ contains $\Pi$ as a cell, corresponding to $x = 0$ under the notation of Theorem 3.3. Since the $\epsilon _{f,k}$ are generic, all other cells of $\mathscr {S}_A$ can be written as $F \cup D$, where F is a face of $\Pi$ and D is a simplex affinely independent to F. Hence, any triangulation of the cell $\Pi$ can be extended to a refinement of $\mathscr {S}_A$ which is a triangulation of A. $\square$

5.3 Construction of a Zonotopal Tiling

We now construct a triangulation of $\Pi$ from which we will obtain our collections $\mathscr {C}_{S,i,j}$. For each $(ij) \in \Gamma _5^2$, partition $\Delta _{(ij)}$ into the following sets: $\Delta _{i,j},\Delta _{j,i}$, and

$$\begin{aligned}&\Delta _{S,i',j',(ij)} \text { for each } S \in {\textstyle \left( {\begin{array}{c}[5]\\ 4\end{array}}\right) }\text { with } i,j, \in S\\&\quad \text { and distinct } i', j' \in S \text { with } \{i',j'\} \ne \{i,j\}. \end{aligned}$$

Next, for each distinct $i,j \in [5]$, choose an element $f_{i,j}^*\in \Delta _{i,j}$. For each $S \in \left( {\begin{array}{c}[5]\\ 4\end{array}}\right)$ with $i,j \in S$, let $\Delta _{S,i,j,(ij)} \subseteq \Delta _{i,j}$ be sets such that

$$\begin{aligned} \Delta _{i,j} = \bigcup _{\begin{array}{c} S \in \left( {\begin{array}{c}[5]\\ 4\end{array}}\right) \\ i,j \in S \end{array}} \Delta _{S,i,j,(ij)} \end{aligned}$$

and

$$\begin{aligned} \Delta _{S,i,j,(ij)} \cap \Delta _{S',i,j,(ij)} = \{f_{i,j}^*\}\quad \hbox { for each distinct}\quad S, S' \in {\textstyle \left( {\begin{array}{c}[5]\\ 4\end{array}}\right) } \hbox {,}\quad i,j \in S, S'. \end{aligned}$$

The sizes of all of these sets will be determined later.

Now, we let $\mathscr {S} {:=} \mathscr {S}_\Pi ^\omega$ be the regular subdivision of $\Pi$ with $\omega :\Pi \rightarrow \mathbb R$ defined as follows. First, for every distinct $i,j \in [5]$, we set

Finally, for each $S \in \left( {\begin{array}{c}[5]\\ 4\end{array}}\right)$ and distinct $i,j,k \in S$, let $0< \epsilon _{S,i,j,k} < 1$ be a generic real number. We set

We analyze the cells of $\mathscr {S}$. For each $S \in \left( {\begin{array}{c}[5]\\ 4\end{array}}\right)$ and distinct $i,j \in S$, let

$$\begin{aligned} \Pi _{S,i,j}\, {:=} \bigcup _{(i'j') \in \Gamma _S^2} \bigcup _{f \in \Delta _{S,i,j,(i'j')}} \{(e_{i'},f),(e_{j'},f)\} \end{aligned}$$

and

$$\begin{aligned} P_{S,i,j}\, {:=} \bigcup _{f \in \Delta _{S,i,j,(ij)}} \{(e_i,f),(e_j,f)\}. \end{aligned}$$

In addition, for each $k \in S \,\backslash \,\{i,j\}$, let

$$\begin{aligned} \Xi _{S,i,j,k}\, {:=} \bigcup _{(i'j') \in \Gamma _{\{i,j,k\}}^2} \bigcup _{f \in \Delta _{S,i,j,(i'j')}} \{(e_{i'},f),(e_{j'},f)\}. \end{aligned}$$

Proposition 5.8

Every cell of $\mathscr {S}$ is of the form $C \cup D$, where D is a simplex affinely independent to C and C is a face of one of the following.

• (a) $\Pi _{S,i,j} \cup \Xi _{S',i,j,k} \cup P_{S'',i,j}$ where $S,S',S'' \in \left( {\begin{array}{c}[5]\\ 4\end{array}}\right)$ are distinct and contain i, j and $\{k\} = [5] \,\backslash \,S$.

• (b) $\Xi _{S,i,j,k} \cup \Xi _{S',i,j,k'} \cup \Xi _{S'',i,j,k''}$ where $S,S',S'' \in \left( {\begin{array}{c}[5]\\ 4\end{array}}\right)$ are distinct and contain i, j and $k \in S \,\backslash \,\{i,j\}$, $k' \in S' \,\backslash \,\{i,j\}$, and $k'' \in S'' \,\backslash \,\{i,j\}$ are distinct.

We will call the cells in (a) and (b) the complex cells of $\mathscr {S}$.

Proof

Using the notation of Theorem 3.3, the cell in (a) corresponds to $x \in \mathbb R^5$ with $x_i = 0$, $x_j = 1$, $x_l = \epsilon _{S,i,j,l}$ for each $l \in S \,\backslash \,\{i,j\}$, and $x_k = \epsilon _{S',i,j,k}$. The cell in (b) corresponds to $x \in \mathbb R^5$ with $x_i = 0$, $x_j = 1$, $x_k = \epsilon _{S,i,j,k}$, $x_{k'} = \epsilon _{S',i,j,k'}$, and $x_{k''} = \epsilon _{S'',i,j,k''}$. Due to the genericness of the $\epsilon _{S,i,j,k}$, it can be checked that every cell of $\mathscr {S}$ is the union of a face of one of these cells and an affinely independent simplex; we leave the details to the reader. $\square$

Thus, in order to give a triangulation of $\Pi$ which refines $\mathscr {S}$, it suffices to specify triangulations of the complex cells of $\mathscr {S}$ which agree on common faces. To do this, we first specify triangulations of each $\Pi _{S,i,j}$. We will then use a “pseudoproduct” operation to extend these to triangulations of the complex cells.

Fix $S \in \left( {\begin{array}{c}[5]\\ 4\end{array}}\right)$ and distinct $i,j \in S$. Let ${\widetilde{\Pi }}$ be the large 3-permutohedron defined in equation (4.3). Let $\psi :[4] \rightarrow S$ be a map such that $\psi (1) = i$ and $\psi (2) = j$. We now choose the sizes of the $\Delta _{S,i,j,(i'j')}$ so that we have an affine isomorphism $\Psi :{\widetilde{\Pi }} \rightarrow \Pi _{S,i,j}$ given by $e_k \mapsto e_{\psi (k)}$ for all $k \in [4]$ and so that $\{f_{kl}^r\}_{-N \le r \le N}$ maps bijectively to $\Delta _{S,i,j,(\psi (k)\psi (l))}$. We will define $\Psi :{\widetilde{\Pi }} \rightarrow \Pi _{S,i,j}$ so that $f_{12}^{-N}$ maps to $f_{i,j}^*$.

Let $\widetilde{\mathscr {T}}$ and $\widetilde{\mathscr {C}}$ be the triangulation of ${\widetilde{\Pi }}$ and the collection of triangulations, respectively, constructed in Sect. 4.4 which satisfy Proposition 4.10. Let $\mathscr {T}_{S,i,j}$ and $\widetilde{\mathscr {C}}_{S,i,j}$ be the images of $\widetilde{\mathscr {T}}$ and $\widetilde{\mathscr {C}}$, respectively, under $\Psi$. We thus have a triangulation $\mathscr {T}_{S,i,j}$ of each $\Pi _{S,i,j}$.

To extend the $\mathscr {T}_{S,i,j}$ to triangulations of the complex cells, we use the following “ordered pseudoproduct” construction. It is proved in the Appendix.

Proposition 5.9

Let $\Pi _1,\Pi _2,\dots , \Pi _N$, and $\rho$ be cells of $\Pi$ such that $\rho = \{(e_i,f),(e_j,f)\}$ for some distinct $i,j \in [5]$ and $f \in \Delta _{(ij)}$, and for any $1 \le r,s \le N$, we have $G(\Pi _r) \cap G(\Pi _s) = G(\sigma )$. Let $\mathscr {T}_1, \dots , \mathscr {T}_N$ be triangulations of $\Pi _1, \dots , \Pi _N$, respectively. Let $\mathscr {M}$ be the set of all simplices $\sigma \in A$ of the following form: There is an integer $1 \le s \le N$ such that

$$\begin{aligned} \sigma = \Big ( \bigcup _{r < s} (\sigma _r \,\backslash \,\rho ) \Big ) \cup \sigma _s \cup \Big ( \bigcup _{r > s} \sigma _r \Big ) \end{aligned}$$

where

• If $r < s$, then $\sigma _r$ is a maximal simplex of $\mathscr {T}_r$ with $\rho \subseteq \sigma _r$.

• $\sigma _s$ is a maximal simplex of $\mathscr {T}_s$, and if $s < N$, then $\rho ' \,{:=}\, \sigma _s \cap \rho \ne \rho$.

• If $r > s$, then $\sigma _r$ is a maximal simplex of $\mathscr {T}_r[F_r]$, where $F_r$ is a facet of $\Pi _r$ with $F_r \cap \rho = \rho '$.

Then $\mathscr {M}$ is the set of maximal simplices of a triangulation $\mathscr {T}(\mathscr {T}_1,\dots ,\mathscr {T}_N)$ of $\Pi _1 \cup \cdots \cup \Pi _N$, and $\mathscr {T}_r \subseteq \mathscr {T}(\mathscr {T}_1,\dots ,\mathscr {T}_N)$ for all $1 \le r \le N$.

Now, fix distinct $i,j \in [5]$, and specify an ordering $S_1,S_2,S_3$ of the sets $S \in \left( {\begin{array}{c}[5]\\ 4\end{array}}\right)$ which contain i and j. Each complex cell is of the form $F_1 \cup F_2 \cup F_3$ where for each distinct $1 \le r,s \le 3$, $F_r$ is a face of $\Pi _{S_r,i,j}$, and $G(F_r) \cap G(F_s) = G(\rho )$ where

$$\begin{aligned} \rho \,{:=}\, \{(e_i,f_{i,j}^*),(e_j,f_{i,j}^*)\}. \end{aligned}$$

By Proposition 5.9, we thus have a triangulation $\mathscr {T}( \mathscr {T}_{S_1,i,j}[F_1], \mathscr {T}_{S_2,i,j}[F_2], \mathscr {T}_{S_3,i,j}[F_3] )$ of $F_1 \cup F_2 \cup F_3$. 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 $\mathscr {T}$ of $\Pi$. By Proposition 5.9, this triangulation $\mathscr {T}$ contains all the triangulations $\mathscr {T}_{S,i,j}$ as subcollections.

For each $S \in \left( {\begin{array}{c}[5]\\ 4\end{array}}\right)$ and distinct $i,j \in [n]$, let $\mathscr {D}_{S,i,j}$ be the set of all zonotopal triangulations of the form $\mathscr {T}_{ijk}^{f_1f_2f_3}$ such that $k \in S \,\backslash \,\{i,j\}$ and $\mathscr {T}_{ijk}^{f_{i,j}^*f_2f_3} \in \widetilde{\mathscr {C}}_{S,i,j}$. Let

$$\begin{aligned} \mathscr {C}_{S,i,j}\, {:=} \,\widetilde{\mathscr {C}}_{S,i,j} \cup \mathscr {D}_{S,i,j}. \end{aligned}$$

We now finally show that the assumptions of Lemma 5.2 hold for $\mathscr {C}_{S,i,j}$ and $\mathscr {T}$.

Proposition 5.10

The collections $\mathscr {C}_{S,i,j}$ satisfy Properties 1–3 of Lemma 5.2.

Proof

We first prove that Properties 1 and 2 hold. Suppose $\mathscr {T}_{i_1i_2i_3}^{f_1f_2f_3} \in \mathscr {C}_{S,i,j}$. If $\mathscr {T}_{i_1i_2i_3}^{f_1f_2f_3} \in \widetilde{\mathscr {C}}_{S,i,j}$, then Properties 1 and 2 follow from Proposition 4.10 (B) by setting the appropriate values of $\gamma$. So we may assume $\mathscr {T}_{i_1i_2i_3}^{f_1f_2f_3} = \mathscr {T}_{ijk}^{f_1f_2f_3} \in \mathscr {D}_{S,i,j}$. By definition of $\mathscr {D}_{S,i,j}$, we have $\mathscr {T}_{ijk}^{f_{i,j}^*f_2f_3} \in \widetilde{\mathscr {C}}_{S,i,j}$. Hence, by Proposition 4.10 (B), there exist $f_1' \in \Delta _{S,i,j,(il)}$, $f_2' \in \Delta _{S,i,j,(jl)}$, and $f_3' \in \Delta _{S,i,j,(kl)}$, where $l = S \,\backslash \,\{i,j,k\}$, such that

$$\begin{aligned} \mathscr {T}_{ijl}^{f_{i,j}^*f_2'f_1'}, \mathscr {T}_{jkl}^{f_2f_3'f_2'}, \mathscr {T}_{kil}^{f_3f_1'f_3'} \in \widetilde{\mathscr {C}}_{S,i,j}. \end{aligned}$$

By definition, we thus have $\mathscr {T}_{ijl}^{f_1f_2'f_1'} \in \mathscr {D}_{S,i,j}$. Hence

$$\begin{aligned} \mathscr {T}_{ijl}^{f_1f_2'f_1'}, \mathscr {T}_{jkl}^{f_2f_3'f_2'}, \mathscr {T}_{kil}^{f_3f_1'f_3'} \in \mathscr {C}_{S,i,j} \end{aligned}$$

which proves Property 1. The argument for Property 2 is analogous.

We now prove Property 3. Let $i,j,l \in [5]$ be distinct and let $f_1 \in \Delta _{(ij)}$. Let $S = [5] \,\backslash \,\{l\}$. By Proposition 4.10 (A) applied to the triangulation $\mathscr {T}_{S,i,j}$, for any $k \in S \,\backslash \,\{i,j\}$ there exists $f_2 \in \Delta _{S,i,j,(jk)}$ and $f_3 \in \Delta _{S,i,j,(ki)}$ such that

$$\begin{aligned} \mathscr {T}_{ijk}^{f_{i,j}^*f_2 f_3} \in \widetilde{\mathscr {C}}_{S,i,j}. \end{aligned}$$

Thus by definition, $\mathscr {T}_{ijk}^{f_1f_2f_3} \in \mathscr {D}_{S,i,j} \subseteq \mathscr {C}_{S,i,j}$, which proves Property 3. $\square$

Proposition 5.11

Properties (i) and (ii) of Lemma 5.2 hold for $\mathscr {T}$ with respect to the collections $\mathscr {C}_{S,i,j}$.

Proof

We first note the following two facts.

Proposition 5.12

For any $f \in \Delta _{(ij)}$, we have $\{ (e_i, f_{i,j}^*), (e_j, f) \} \in \mathscr {T}$.

Proof

In our construction of $\mathscr {S}$ from $\omega$, we had

$$\begin{aligned} \omega (e_i,f_{i,j}^*) - \omega (e_j,f_{i,j}^*) + \omega (e_j,f) - \omega (e_i,f) \le 0 \end{aligned}$$

for all $f \in \Delta _{(ij)}$, with equality if and only if $f \in \Delta _{i,j}$. In addition, $\widetilde{\mathscr {T}}$ is a refinement of a regular subdivision of $\widetilde{\Pi }$ given by height function ${\widetilde{\omega }}$ where

$$\begin{aligned} {\widetilde{\omega }}(e_1,f_{12}^{-N}) - {\widetilde{\omega }}(e_2,f_{12}^{-N}) + {\widetilde{\omega }}(e_2,f_{12}^r) - {\widetilde{\omega }}(e_1,f_{12}^r) < 0 \end{aligned}$$

for all $r \ne N$. Thus, for all $f \in \Delta _{(ij)}$, restricting $\mathscr {T}$ to the face $X_{ij}^{f_{i,j}^*f}$ of $\Pi$ yields the triangulation $\mathscr {T}_{X_{ij}^{f_{i,j}^*f}}^-$, and hence $\{ (e_i, f_{i,j}^*), (e_j, f) \} \in \mathscr {T}$. $\square$

Proposition 5.13

Let $\mathscr {T}_{ijk}^{f_{i,j}^*f_2f_3} \subseteq \mathscr {T}$ and $f_1 \in \Delta _{(ij)}$. Then $\mathscr {T}_{ijk}^{f_1f_2f_3} \subseteq \mathscr {T}$.

Proof

Let

$$\begin{aligned} \sigma _0 \,{:=}\, X_{ijk}^{f_{i,j}^*f_2f_3} \,\backslash \,\{(e_i,f_{i,j}^*)\} \in \mathscr {T}. \end{aligned}$$

By considering a maximal simplex of $\mathscr {T}$ containing $\sigma _0$, we have $\sigma _0 \cup \{(e,f_1)\} \in \mathscr {T}$ for some $e \in \{e_i,e_j\}$. If $e = e_i$, then $\{(e_i,f_1),(e_j,f_{i,j}^*)\} \subseteq \sigma _0$. However, this contradicts Propositions 5.12 and 6.2 in the Appendix. Thus $e = e_j$. Hence, we have

$$\begin{aligned} \sigma _1 \,{:=}\, X_{ijk}^{f_1f_2f_3} \,\backslash \,\{(e_i,f_1)\} \in \mathscr {T}. \end{aligned}$$

Now, the circuit $X\, {:=}\, X_{ijk}^{f_1f_2f_3}$ is a face of $\Pi$, so $\mathscr {T}[X]$ is a triangulation of X. Since $\sigma _1 \in \mathscr {T}[X]$, this triangulation must be $\mathscr {T}_X^+$. Thus $\mathscr {T}_{ijk}^{f_1f_2f_3} \subseteq \mathscr {T}$. $\square$

Now, for any $\mathscr {T}_{i_1i_2i_3}^{f_1f_2f_3} \in \widetilde{\mathscr {C}}_{S,i,j}$, by definition of $\widetilde{\mathscr {T}}$ and $\widetilde{\mathscr {C}}$ we have $\mathscr {T}_{i_1i_2i_3}^{f_1f_2f_3} \subseteq \mathscr {T}_{S,i,j} \subseteq \mathscr {T}$. Suppose $\mathscr {T}_{ijk}^{f_1f_2f_3} \in \mathscr {D}_{S,i,j}$. Then $\mathscr {T}_{ijk}^{f_{i,j}^*f_2f_3} \in \widetilde{\mathscr {C}}_{S,i,j}$ by definition. Thus $\mathscr {T}_{ijk}^{f_{i,j}^*f_2f_3} \subseteq \mathscr {T}$, so by Proposition 5.13, $\mathscr {T}_{ijk}^{f_1f_2f_3} \subseteq \mathscr {T}$. Hence Property (i) holds.

Now suppose $\mathscr {T}_{ijk}^{f_1f_2f_3} \in \mathscr {C}_{S,i,j}$, and let $f \in \Delta _{(il)}$ where $\{l\} = [5] \,\backslash \,S$. By Property (i), we have $X_{ijk}^{f_1f_2f_3} \,\backslash \,\{(e_i,f_1)\} \in \mathscr {T}$. By Proposition 6.1 in the Appendix, we have $X_{ijk}^{f_1f_2f_3} \,\backslash \,\{(e_i,f_1)\} \cup \{(e_i,f)\} \in \mathscr {T}$. Thus Property (ii) holds. $\square$

Hence, we have collections $\mathscr {C}_{S,i,j}$ which satisfy the hypotheses of Lemma 5.2, and by Propositions 5.11 and 5.7, there exists a triangulation $\mathscr {T}' \in \mathcal S_{\mathscr {C}}$. Clearly $\mathcal S_{\mathscr {C}}$ 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 [2]. Hence Lemma 5.2 proves Theorem 5.1.

Appendix

1.1 Proof of Proposition 5.3

Since $\sigma \notin \mathscr {T}'$, by Proposition 2.2 we have $\sigma \supseteq X^-$. Let $\tau$ be a maximal simplex of $\mathscr {T}$ containing $\sigma$. We must have $\tau \notin \mathscr {T}'$ since $\sigma \notin \mathscr {T}'$. So by Proposition 2.2, $\tau$ must contain a maximal simplex $\tau '$ of $\mathscr {T}_X^+$. Now, since $\sigma \cup \tau ' \subseteq \tau$, we have that $G(\sigma \cup \tau ')$ is acyclic. But since $\sigma \supseteq X^-$ and $G(\sigma )$ is connected, this can only happen if $\sigma \supseteq \tau '$, as desired.

1.2 Proof of Proposition 5.4

Throughout this section, let $A \,{:=}\, \Delta ^{m-1} \times \Delta ^{n-1}$. We need the following two facts.

Proposition 6.1

Let $\mathscr {T}$ be a triangulation of A. Let $\sigma \in \mathscr {T}$. Suppose that $f \in \Delta ^{n-1}$ and $f \notin G(\sigma )$. Then there exists $e \in \Delta ^{m-1} \cap G(\sigma )$ such that $\sigma \cup \{(e,f)\} \in \mathscr {T}$.

Proof

Let $\Delta _\sigma \,{:=}\, \Delta ^{m-1} \cap G(\sigma )$. Let $\phi :A \rightarrow \mathbb R$ be the linear functional with

$$\begin{aligned} \phi (e,0)&= {\left\{ \begin{array}{ll} -1 &{} \mathrm{{if}} \quad e \in \Delta _\sigma , \\ 0 &{} \mathrm{{otherwise}}. \end{array}\right. } \\ \phi (0,f)&= \ \; \; 0 \quad \text { for all } f \in \Delta ^{n-1}. \end{aligned}$$

Let F be the face of A which minimizes $\phi$. Then $\sigma \subseteq F$. Let $\tau$ be a maximal simplex of $\mathscr {T}[F]$ containing $\sigma$. Then $\tau \supseteq \sigma \cup \{(e,f)\}$ for some $e \in \Delta _\sigma$. $\square$

By symmetry of $\Delta ^{m-1}$ and $\Delta ^{n-1}$, the above proposition also holds after replacing $f \in \Delta ^{n-1}$ with $e \in \Delta ^{m-1}$.

Proposition 6.2

Let $\mathscr {T}$ be a triangulation of A, and suppose $X = (X^+,X^-)$ is a circuit of A. Then $\mathscr {T}$ does not contain both $X^+$ and $X^-$.

Proof

The interiors of the convex hulls of opposite parts of a circuit intersect. $\square$

The next proposition immediately implies Proposition 5.4.

Proposition 6.3

Let $\mathscr {T}$ be a triangulation of A. Let $i,j,k,l \in [m]$ be distinct , and for each $\alpha \in \Gamma _{\{i,j,k,l\}}^2$, let $f_\alpha \in \Delta ^{n-1}$. Suppose that

$$\begin{aligned} \mathscr {T}_{jkl}^{f_{(jk)}f_{(kl)}f_{(lj)}}, \mathscr {T}_{kil}^{f_{(ki)}f_{(il)}f_{(lk)}} \subseteq \mathscr {T} \end{aligned}$$

and

$$\begin{aligned} X_{ijl}^{f_{(ij)}f_{(jl)}f_{(li)}} \,\backslash \,\{ (e_i, f_{(ij)}) \} \in \mathscr {T}. \end{aligned}$$

Then $\mathscr {T}$ does not have a flip supported on $X_{ijk}^{f_{(ij)}f_{(jk)}f_{(ki)}}$.

Proof

Suppose the contrary. In particular, this means

$$\begin{aligned} \mathscr {T}_{ijk}^{f_{(ij)}f_{(jk)}f_{(ki)}} \subseteq \mathscr {T}. \end{aligned}$$

Set $\sigma _0 = X_{ijl}^{f_{(ij)}f_{(jl)}f_{(li)}} \,\backslash \,\{ (e_i, f_{(ij)}) \}$ (Fig. 3). By assumption, $\sigma _0 \in \mathscr {T}$. By Proposition 6.1, there exists $\sigma _1 \in \mathscr {T}$ such that

$$\begin{aligned} \sigma _1 = \sigma _0 \cup \{(e,f_{(jk)})\} \cup \{(e',f_{(kl)})\}, \end{aligned}$$

where $e,e' \in \{e_i,e_j,e_l\}$. We prove the following claims.

Fig. 3

$G(\sigma _0)$

Claim 1: $e = e_j$.

Suppose first that $e = e_i$. Then $\{(e_i,f_{(jk)}),(e_j,f_{(ij)})\} \subseteq \sigma _1$. However, we also have $\{(e_i,f_{(ij)}),(e_j,f_{(jk)})\} \in \mathscr {T}_{ijk}^{f_{(ij)}f_{(jk)}f_{(ki)}} \subseteq \mathscr {T}$ by assumption. Thus $\mathscr {T}$ contains both parts of the circuit $X_{ij}^{f_{(jk)}f_{(ij)}}$, which contradicts Proposition 6.2.

Now suppose that $e = e_l$. Then $\{(e_l,f_{(jk)}),(e_j,f_{(jl)})\} \subseteq \sigma _1$. However, we also have $\{(e_l,f_{(jl)}),(e_j,f_{(jk)})\} \in \mathscr {T}_{jkl}^{f_{(jk)}f_{(kl)}f_{(lj)}} \subseteq \mathscr {T}$, again a contradiction. So $e = e_j$.

Claim 2: $e' = e_l$.

Suppose first that $e' = e_i$. Then $\{(e_i,f_{(kl)}),(e_l,f_{(li)})\} \subseteq \sigma _1$. However, we have $\{(e_i,f_{(li)}),(e_l,f_{(kl)})\} \in \mathscr {T}_{kil}^{f_{(ki)}f_{(il)}f_{(lk)}} \subseteq \mathscr {T}$, a contradiction.

Now suppose that $e' = e_j$. Then $\{(e_j,f_{(kl)}),(e_l,f_{(jl)})\} \subseteq \sigma _1$. However, we have $\{(e_j,f_{(jl)}),(e_l,f_{(kl)})\} \in \mathscr {T}_{jkl}^{f_{(jk)}f_{(kl)}f_{(lj)}} \subseteq \mathscr {T}$, a contradiction. So $e' = e_l$ (Fig. 4).

Fig. 4

$G(\sigma _1)$

Now, by Proposition 6.1 and the comment afterwards, there exists $\sigma _2 \in \mathscr {T}$ such that

$$\begin{aligned} \sigma _2 = \sigma _1 \cup \{ (e_k,f) \} \end{aligned}$$

where $f \in \{f_{(ij)},f_{(jk)},f_{(il)},f_{(jl)},f_{(kl)}\}$. We prove the following.

Claim 3: $f = f_{(jk)}$.

The argument goes as follows:

If $f = f_{(ij)}$, then $\{(e_k,f_{(ij)}),(e_j,f_{(jk)})\} \subseteq \sigma _2$ but $\{(e_k,f_{(jk)}),(e_j,f_{(ij)})\} \in \mathscr {T}_{ijk}^{f_{(ij)}f_{(jk)}f_{(ki)}}$.

If $f = f_{(il)}$, then $\{(e_k,f_{(il)}),(e_l,f_{(kl)})\} \subseteq \sigma _2$ but $\{(e_k,f_{(kl)}),(e_l,f_{(il)})\} \in \mathscr {T}_{kil}^{f_{(ki)}f_{(il)}f_{(lk)}}$.

If $f = f_{(jl)}$, then $\{(e_k,f_{(jl)}),(e_j,f_{(jk)})\} \subseteq \sigma _2$ but $\{(e_k,f_{(jk)}),(e_j,f_{(jl)})\} \in \mathscr {T}_{jkl}^{f_{(jk)}f_{(kl)}f_{(lj)}}$.

If $f = f_{(kl)}$, then $(X_{jkl}^{f_{(jk)}f_{(kl)}f_{(lj)}})^+ \subseteq \sigma _2$, but $(X_{jkl}^{f_{(jk)}f_{(kl)}f_{(lj)}})^- \in \mathscr {T}_{jkl}^{f_{(jk)}f_{(kl)}f_{(lj)}}$.

Thus $f = f_{(jk)}$ (Fig. 5).

Fig. 5

$G(\sigma _2)$

Finally, by Proposition 6.1 there exists $\sigma _3 \in \mathscr {T}$ such that

$$\begin{aligned} \sigma _3 = \sigma _2 \cup \{(e,f_{(ki)})\} \end{aligned}$$

where $e \in \{e_i,e_j,e_k,e_l\}$.

Claim 4: $e = e_i$.

The argument goes as follows.

If $e = e_j$, then $\{(e_j,f_{(ki)}),(e_k,f_{(jk)})\} \subseteq \sigma _3$ but $\{(e_j,f_{(jk)}),(e_k,f_{(ki)})\} \in \mathscr {T}_{ijk}^{f_{(ij)}f_{(jk)}f_{(ki)}}$.

If $e = e_k$, then $(X_{kil}^{f_{(ki)}f_{(il)}f_{(lk)}})^+ \subseteq \sigma _3$ but $(X_{kil}^{f_{(ki)}f_{(il)}f_{(lk)}})^- \in \mathscr {T}_{kil}^{f_{(ki)}f_{(il)}f_{(lk)}}$.

If $e = e_l$, then $\{(e_l,f_{(ki)}),(e_i,f_{(il)})\} \subseteq \sigma _3$ but $\{(e_l,f_{(il)}),(e_i,f_{(ki)})\} \in \mathscr {T}_{kil}^{f_{(ki)}f_{(il)}f_{(lk)}}$.

Thus $e = e_i$ (Fig. 6).

Fig. 6

$G(\sigma _3)$

Now, let $\tau$ be a maximal simplex of $\mathscr {T}$ containing $\sigma _3$. Then $\bigl (X_{ijk}^{f_{(ij)}f_{(jk)}f_{(ki)}}\bigr ){}^- \subseteq \tau$ but $\bigl |X_{ijk}^{f_{(ij)}f_{(jk)}f_{(ki)}} \cap \tau \bigr | = 4$. This contradicts Proposition 2.3 and our assumption that $\mathscr {T}$ has a flip supported on $X_{ijk}^{f_{(ij)}f_{(jk)}f_{(ki)}}$. So $\mathscr {T}$ does not have such a flip. $\square$

1.3 Proof of Proposition 5.9

Our proof is based on the following.

Theorem 6.4

(Rambau [8]) Let $\mathscr {M}$ be a nonempty collection of full-dimensional simplices of a point set A. Then $\mathscr {M}$ is the set of maximal simplices of a triangulation of A if and only if

• 1. There is no circuit $X = (X^+,X^-)$ of A and simplices $\tau ,\tau ' \in \mathscr {M}$ such that $X^+ \subseteq \tau$, $X^- \subseteq \tau '$.

• 2. For any simplex $\tau \in \mathscr {M}$ and facet $\sigma$ of $\tau$, either $\sigma$ is contained in a facet of A or there is another $\tau ' \in \mathscr {M}$, $\tau ' \ne \tau$, such that $\sigma \subseteq \tau '$.

Let $\mathscr {M}$ be as in Proposition 5.9. We prove Properties 1 and 2 of Theorem 6.4 for $\mathscr {M}$.

Proof of Property 1

Suppose that $X = (X^+,X^-)$ is a circuit of $\Pi _1 \cup \cdots \cup \Pi _N$ and we have $X^+ \subseteq \sigma$, $X^- \subseteq \sigma '$ for some $\sigma , \sigma ' \in \mathscr {M}$. Let

$$\begin{aligned} \sigma&= \Big ( \bigcup _{r< s} (\sigma _r \,\backslash \,\rho ) \Big ) \cup \sigma _s \cup \Big ( \bigcup _{r> s} \sigma _r \Big ), \\ \sigma '&= \Big ( \bigcup _{r < s'} (\sigma '_r \,\backslash \,\rho ) \Big ) \cup \sigma '_{s'} \cup \Big ( \bigcup _{r > s'} \sigma '_r \Big ) \end{aligned}$$

as in Proposition 5.9, with analogous definitions for $\sigma '$. Suppose first that $X \subseteq \Pi _r$ for some $1 \le r \le N$. Then we have $X^+ \subseteq \sigma \cap \Pi _r \subseteq \sigma _r$ and similarly $X^- \subseteq \sigma '_r$. This contradicts the fact that $\sigma _r,\sigma '_r$ are simplices of a triangulation $\mathscr {T}_r$. So we cannot have this case. It follows that $X \subseteq \Pi _{r_1} \cup \Pi _{r_2}$ for some $1 \le r_1 < r_2 \le N$, and we can uniquely write

$$\begin{aligned} X = X_{r_1} \cup X_{r_2} \,\backslash \,\rho , \end{aligned}$$

where $X_{r_1} = (X_{r_1}^+,X_{r_1}^-)$ and $X_{r_2} = (X_{r_2}^+,X_{r_2}^-)$ are circuits in $\Pi _{r_1}$ and $\Pi _{r_2}$ respectively, and

where $\rho _1,\rho _2$ are different one-element subsets of $\rho$.

We claim that $X_{r_1}^+ \subseteq \sigma _{r_1}$. First suppose that $r_1 < s$. Then $(X^+ \cap \Pi _{r_1}) \cup \rho \subseteq \sigma _{r_1}$, so $X_{r_1}^+ \subseteq \sigma _{r_1}$, as desired. So we may assume $r_1 \ge s$, and hence $s < r_2 \le N$. Now, since $s < N$, we have $\sigma \cap \rho = \rho _i$ for either $i = 1$ or 2. If $i = 1$, then $X_{r_1}^+ \subseteq \sigma _{r_1}$ and we are done. Suppose $i = 2$. Then $X_{r_2}^+ \subseteq \sigma _{r_2}$. Since $r_2 > s$, it follows that $X_{r_2}^+$ is contained in a facet F of $\Pi _{r_2}$ with $F \cap \rho = \rho _2$. However, $X_{r_2}^+ \subseteq F$ implies $X_{r_2} \subseteq F$ because F is a face of $\Pi _{r_2}$. This contradicts $F \cap \rho = \rho _2$. Hence we have $X_{r_1}^+ \subseteq \sigma _{r_1}$.

By the same argument, we have $X_{r_1}^- \subseteq \sigma '_{r_1}$. Hence $X_{r_1}^+,X_{r_1}^- \in \mathscr {T}_{r_1}$, a contradiction. This proves Property 1. $\square$

Proof of Property 2

Let

$$\begin{aligned} \sigma = \Big ( \bigcup _{r < s} (\sigma _r \,\backslash \,\rho ) \Big ) \cup \sigma _s \cup \Big ( \bigcup _{r > s} \sigma _r \Big ) \end{aligned}$$

be an element of $\mathscr {M}$ as before. Let $x \in \sigma$, and consider the facet $\sigma \,\backslash \,\{x\}$. We have the following cases.

Case 1: $x \in \sigma _t \,\backslash \,\rho$ where $t < s$.

Subcase 1.1: $\sigma _t \,\backslash \,\{x\}$ is not contained in a facet of $\Pi _t$. Then there is some $x' \in \Pi _t$ such that $\sigma _t' \,{:=}\, \sigma _t \,\backslash \,\{x\} \cup \{x'\} \in \mathscr {T}_t$. Then

$$\begin{aligned} \sigma \,\backslash \,\{x\} \cup \{x'\} = \Big ( \bigcup _{\begin{array}{c} r < s \\ r \ne t \end{array}} (\sigma _r \,\backslash \,\rho ) \Big ) \cup (\sigma _t' \,\backslash \,\rho ) \cup \sigma _s \cup \Big ( \bigcup _{r > s} \sigma _r \Big ) \end{aligned}$$

is an element of $\mathscr {M}$ containing $\sigma \,\backslash \,\{x\}$, as desired.

Subcase 1.2: $\sigma _t \,\backslash \,\{x\}$ is contained in a facet F of $\Pi _t$. Let $\phi :\Pi _t \rightarrow \mathbb R$ be a linear functional supporting F on $\Pi _t$. Since $\rho \subseteq \sigma _t \,\backslash \,\{x\}$, we have $\rho \subseteq F$, hence $\phi (\rho ) = \{b\}$ where b is the minimum of $\phi$ on $\Pi _t$. We may assume $b = 0$.² Now, we can extend $\phi$ to a linear functional $\phi '$ on $\Pi _1 \cup \cdots \cup \Pi _N$ by setting $\phi '(v) = \phi (v)$ for all $v \in \Pi _t$ and $\phi '(v) = 0$ for all $v \in \Pi _r$, $r \ne t$.³ Then the face of $\Pi _1 \cup \cdots \cup \Pi _N$ supported by $\phi '$ contains $\sigma \,\backslash \,\{x\}$, and this face is proper because it does not contain $\Pi _t$. Thus $\sigma \,\backslash \,\{x\}$ is contained in a facet of $\Pi _1 \cup \cdots \cup \Pi _N$, as desired.

Case 2: $x \in \rho$.

Subcase 2.1: $\sigma \cap \rho \ne \rho$. Then $\sigma \,\backslash \,\{x\}$ is contained in a facet of $\Pi _1 \cup \cdots \cup \Pi _N$ because f has no neighbors in $G(\sigma \,\backslash \,\{x\})$.

Subcase 2.2: $\sigma \cap \rho = \rho$. This implies $s = N$ and $x \in \sigma _r$ for all r. First, suppose there is some t such that $\sigma _t \,\backslash \,\{x\}$ is not contained in a facet of $\Pi _t$, and let t be the largest such number. Then there is some $x' \in \Pi _t$ such that $\sigma _t'\, {:=}\, \sigma _t \,\backslash \,\{x\} \cup \{x'\} \in \mathscr {T}_t$. Thus

$$\begin{aligned} \sigma \,\backslash \,\{x\} \cup \{x'\} = \Big ( \bigcup _{r < t} (\sigma _r \,\backslash \,\rho ) \Big ) \cup \sigma _t' \cup \Big ( \bigcup _{r > t} (\sigma _r \,\backslash \,\{x\}) \Big ) \end{aligned}$$

is an element of $\mathscr {M}$ containing $\sigma \,\backslash \,\{x\}$, as desired.⁴

Now suppose there is no such t. Then for all $r,\sigma _r \,\backslash \,\{x\}$ is contained in a facet $F_r$ of $\Pi _r$. We have $x \notin F_r$ since $\sigma _r$ is full dimensional in $\Pi _r$. For each r, let $\phi _r :\Pi _r \rightarrow \mathbb R$ be a linear functional supporting $F_r$ on $\Pi _r$. Let $\{y\} = \rho \,\backslash \,\{x\}$. Then $y \in F_r$ and $x \notin F_r$, so $\phi _r(y) < \phi _r(x)$. By appropriately choosing $\phi _r$, we may assume $\phi _r(y) = 0$ and $\phi _r(x) = 1$ for all r. As before, we can then define a linear functional $\phi '$ on $\Pi _1 \cup \cdots \cup \Pi _N$ such that $\phi '(v) = \phi _r(v)$ for all $v \in \Pi _r$. Then the face of $\Pi _1 \cup \cdots \cup \Pi _N$ supported by $\phi '$ contains $\sigma \,\backslash \,\{x\}$, and this face is proper. Thus $\sigma \,\backslash \,\{x\}$ is contained in a facet of $\Pi _1 \cup \cdots \cup \Pi _N$.

Case 3: $x \in \sigma _s \,\backslash \,\rho$.

Subcase 3.1: $\sigma _s \,\backslash \,\{x\}$ is not contained in a facet of $\Pi _s$. Then there is some $x' \in \Pi _s$ such that $\sigma '_s \,{:=}\, \sigma _s \,\backslash \,\{x\} \cup \{x'\} \in \mathscr {T}_s$. First assume that $x' \notin \rho$. Then

$$\begin{aligned} \sigma \,\backslash \,\{x\} \cup \{x'\} = \Big ( \bigcup _{r < s} (\sigma _r \,\backslash \,\rho ) \Big ) \cup \sigma _s' \cup \Big ( \bigcup _{r > s} \sigma _r \Big ) \end{aligned}$$

is an element of $\mathscr {M}$ containing $\sigma \,\backslash \,\{x\}$, as desired.

Now assume that $x' \in \rho$. First, suppose that for all $r > s$, $\sigma _r' \,{:=}\, \sigma _r \cup \{x'\} \in \mathscr {T}_r$. Then

$$\begin{aligned} \sigma \,\backslash \,\{x\} \cup \{x'\} = \Big ( \bigcup _{r< s} (\sigma _r \,\backslash \,\rho ) \Big ) \cup (\sigma _s' \,\backslash \,\rho ) \cup \Big ( \bigcup _{s< r < N} (\sigma _r' \,\backslash \,\rho ) \Big ) \cup \sigma _N' \end{aligned}$$

is an element of $\mathscr {M}$ containing $\sigma \,\backslash \,\{x\}$, as desired. Now suppose that there is some $t > s$ such that $\sigma _t \cup \{x'\} \notin \mathscr {T}_t$, and let t be the smallest such number. Let $x''$ be a point in $\mathscr {T}_t$ such that $x'' \notin \sigma _t$ and $\sigma _t'' \,{:=}\, \sigma _t \cup \{x''\} \in \mathscr {T}_t$. By definition of t, $x'' \ne x'$, and hence $x'' \notin \rho$. Then

$$\begin{aligned} \sigma \,\backslash \,\{x\} \cup \{x''\} = \Big ( \bigcup _{r< s} (\sigma _r \,\backslash \,\rho ) \Big ) \cup (\sigma _s' \,\backslash \,\rho ) \cup \Big ( \bigcup _{s< r < t} (\sigma _r' \,\backslash \,\rho ) \Big ) \cup \sigma _t'' \cup \Big ( \bigcup _{r > t} \sigma _r \Big ) \end{aligned}$$

is an element of $\mathscr {M}$ containing $\sigma \,\backslash \,\{x\}$, as desired.

Subcase 3.2: $\sigma _s \,\backslash \,\{x\}$ is contained in a facet F of $\Pi _s$. If $\rho \subseteq F$, then by the same argument as in Subcase 1.2, $\sigma \,\backslash \,\{x\}$ is contained in a facet of $\Pi _1 \cup \cdots \cup \Pi _N$. So we may assume $\rho \not \subseteq F$. In particular, this means $\sigma _s \cap \rho = \rho \,\backslash \,\{x'\}$ for some $x' \in \rho$.

First, suppose that there is some $t < s$ such that $\sigma _t \,\backslash \,\{x'\}$ is not contained in a facet of $\mathscr {T}_t$, and let t be the largest such number. Then there is some $x'' \in \mathscr {T}_t$ such that $\sigma _t'' \,{:=}\, \sigma _t \,\backslash \,\{x'\} \cup \{x''\} \in \mathscr {T}_t$. Then

$$\begin{aligned} \sigma \,\backslash \,\{x\} \cup \{x''\} = \Big ( \bigcup _{r< t} (\sigma _r \,\backslash \,\rho ) \Big ) \cup \sigma _t'' \cup \Big ( \bigcup _{t< r < s} (\sigma _r \,\backslash \,\{x'\}) \Big ) \cup (\sigma _s \,\backslash \,\{x\}) \cup \Big ( \bigcup _{r > s} \sigma _r \Big ) \end{aligned}$$

is an element of $\mathscr {M}$ containing $\sigma \,\backslash \,\{x\}$, as desired.

Now suppose that there is no such t. Then for all r, we have that $(\sigma \,\backslash \,\{x\}) \cap \Pi _r$ is contained in a facet $F_r$ of $\Pi _r$. We also have $F_r \cap \rho = \rho \,\backslash \,\{x'\}$ for all r: For $r < s$, this holds because $\sigma _r$ is maximal in $\Pi _r$, for $r = s$, this holds by our original assumption, and for $r > s$, this holds by definition. Hence, by the argument in the second part of Subcase 2.2, $\sigma \,\backslash \,\{x\}$ is contained in a facet of $\Pi _1 \cup \cdots \cup \Pi _N$, as desired.

Case 4: $x \in \sigma _t \,\backslash \,\rho$ where $t > s$.

Let $\rho ' = \sigma _t \cap \rho$. Let $\mathscr {U} \subseteq \mathscr {T}_t$ be the simplicial complex consisting of all simplices in $\mathscr {T}_t$ on the boundary of $\Pi _t$. Let $\mathscr {U}'$ be the subcomplex of $\mathscr {U}$ consisting of all simplices in $\mathscr {T}_t$ contained in facets F of $\Pi _t$ with $F \cap \rho = \rho '$. In particular, we have $\sigma _t \in \mathscr {U}'$.

Since $\mathscr {U}$ is homeomorphic to a sphere, there is some $x' \in \Pi _t$ such that $\sigma _t' \,{:=}\, \sigma _t \,\backslash \,\{x\} \cup \{x'\} \in \mathscr {U}$. If $\sigma _t' \in \mathscr {U}'$, then

$$\begin{aligned} \sigma \,\backslash \,\{x\} \cup \{x'\} = \Big ( \bigcup _{r < s} (\sigma _r \,\backslash \,\rho ) \Big ) \cup \sigma _s \cup \sigma _t' \cup \Big ( \bigcup _{\begin{array}{c} r > s \\ r \ne t \end{array}} \sigma _r \Big ) \end{aligned}$$

is an element of $\mathscr {M}$ containing $\sigma \,\backslash \,\{x\}$, as desired. Suppose $\sigma _t' \notin \mathscr {U}'$. Let F be the facet of $\Pi _t$ containing $\sigma _t'$. Since $\rho ' \subseteq \sigma _t'$, but $F \cap \rho \ne \rho '$, we must have $\rho \subseteq F$. By the argument from Subcase 1.2, it follows that $\sigma \,\backslash \,\{x\}$ is contained in a facet of $\Pi _1 \cup \cdots \cup \Pi _N$, as desired. $\square$

This proves that $\mathscr {M}$ is the set of maximal simplices of a triangulation $\mathscr {T}$. Suppose that $\sigma _s$ is a maximal simplex of $\mathscr {T}_s$. If $\rho \not \subseteq \sigma _s$, then for all $r \ne s$ we can choose $\sigma _r \in \Pi _r$ such that

$$\begin{aligned} \Big ( \bigcup _{r < s} (\sigma _r \,\backslash \,\rho ) \Big ) \cup \sigma _s \cup \Big ( \bigcup _{r > s} \sigma _r \Big ) \end{aligned}$$

is in $\mathscr {M}$. If $\rho \subseteq \sigma _s$, then for all $r \ne s$ we can choose $\sigma _r \in \Pi _r$ such that $\rho \subseteq \sigma _r$ for all r and

$$\begin{aligned} \sigma = \Big ( \bigcup _{r< s} (\sigma _r \,\backslash \,\rho ) \Big ) \cup (\sigma _s \,\backslash \,\rho ) \cup \Big ( \bigcup _{s< r < N} (\sigma _r \,\backslash \,\rho ) \Big ) \cup \sigma _N \end{aligned}$$

is in $\mathscr {M}$. Either way, $\sigma _s \in \mathscr {T}$. Thus, $\mathscr {T}_1, \dots , \mathscr {T}_N \subseteq \mathscr {T}$.