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Published online by Cambridge University Press:  25 October 2021

School of Mathematics, Statistics and Physics, Newcastle University, Newcastle upon Tyne NE1 7RU, UK
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Using a result of Vdovina, we may associate to each complete connected bipartite graph $\kappa $ a two-dimensional square complex, which we call a tile complex, whose link at each vertex is $\kappa $ . We regard the tile complex in two different ways, each having a different structure as a $2$ -rank graph. To each $2$ -rank graph is associated a universal $C^{\star }$ -algebra, for which we compute the K-theory, thus providing a new infinite collection of $2$ -rank graph algebras with explicit K-groups. We determine the homology of the tile complexes and give generalisations of the procedures to complexes and systems consisting of polygons with a higher number of sides.

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1 Introduction

In [Reference Vdovina15], it was shown how to construct a two-dimensional CW-complex whose link at each vertex is a complete bipartite graph. In [Reference Kumjian and Pask6], generalising the work of [Reference Robertson and Steger10], certain combinatorial objects called higher-rank graphs were defined and then associated with a generalisation of a graph algebra [Reference Raeburn9, Ch. 1]. We combine these two methods to build an infinite family of $C^{\star }$ -algebras corresponding to complete bipartite graphs.

We begin in Section 2 by detailing Vdovina’s construction of the CW-complexes, called tile complexes; the data we use to build these is called a tile system. In Sections 3 and 5, we associate adjacency matrices to the tile systems in two different ways: by considering the tiles as pointed and as unpointed geometrical objects, as in [Reference Konter and Vdovina5]. Since these adjacency matrices commute, they characterise the structure of a higher-rank graph, and induce a universal $C^{\star }$ -algebra: the higher-rank graph algebra. We use a result of [Reference Evans2] to calculate the K-groups of these algebras (Theorems 3.10 and 5.3).

In Section 6, we show that the tile complexes have torsion-free homology groups given by $H_1 \cong \mathbb{Z}^{\alpha + \beta + 2},\ H_2 \cong \mathbb{Z}^{(\alpha - 1)(\beta - 1)}$ and $H_n = 0$ otherwise.

Finally, we explore extensions of these methods to $2t$ -gon systems, constructed analogously from two-dimensional complexes consisting entirely of $2t$ -gons. In all, we associate $2$ -rank graph $C^{\star }$ -algebras to five systems, and compute their K-theory in the following theorems:

  1. (1) pointed and unpointed tile systems (Theorems 3.10 and 5.3);

  2. (2) pointed and unpointed $2t$ -gon systems for even t (Theorem 7.4 and Corollary 7.6);

  3. (3) pointed $2t$ -gon systems for arbitrary t (Theorem 7.11).

The respective systems in (2) directly generalise those in (1); however, there is another intuitive way of building $2t$ -gon systems from polyhedra, (3). We discuss the naturality of these generalisations in Section 7.

Our approach differs from that of Robertson and Steger [Reference Robertson and Steger10], who focused on complexes with one vertex. Furthermore, we use the terminology of higher-rank graphs in order to demonstrate the large intersection between the fields of k-graphs and geometry.

Throughout this paper, $\alpha $ , $\beta $ are positive integers, and $\kappa (\alpha ,\beta )$ denotes the complete connected bipartite graph on $\alpha $ white and $\beta $ black vertices.

2 The tile system associated to a bipartite graph

Definition 2.1. Let $t \in \mathbb {Z}$ satisfy $t \geq 2$ , and let $A_1,\ldots ,A_n$ be a sequence of solid t-gons, with directed edges labelled from some set $\mathcal {U}$ . By gluing together like-labelled edges (respecting their direction), we obtain a two-dimensional complex P. We call such a complex a t-polyhedron.

The link at a vertex z of P is the graph obtained as the intersection of P with a small $2$ -sphere centred at z.

Theorem 2.2 (Vdovina [Reference Vdovina15])

Let G be a connected bipartite (undirected) graph on $\alpha $ white and $\beta $ black vertices, with edge set $E(G)$ . Then, for each $t \geq 1$ , we can construct a $2t$ -polyhedron $P(G)$ which has G as the link at each vertex.

We refer to [Reference Vdovina15], in which it was shown how to build such a $2t$ -polyhedron. The general method is as follows.

Write $U'=\lbrace u_1 , \ldots , u_\alpha \rbrace $ for the set of white vertices of G, and $V'=\lbrace v_1,\ldots , v_\beta \rbrace $ for the set of black vertices.

Let U be a set with $2t\alpha $ elements, indexed $u_i^1, u_i^2, \ldots , u_i^t, \bar {u}_i^1, \bar {u}_i^2,\ldots \bar {u}_i^t$ for each $u_i \in U'$ , and let V be the corresponding set with $2t\beta $ elements. Define fixed-point-free involutions $u_i^r \mapsto \bar {u}_i^r$ and $v_i^r \mapsto \bar {v}_i^r$ in U and V, respectively.

Each edge of the graph G joins an element of $U'$ to an element of $V'$ ; for each edge $e = u_p v_q$ , we construct a $2t$ -gon $A_e$ with a distinguished base vertex. Label the boundary of $A_e$ anticlockwise, starting from the base, by the sequence $u_p^1,v_q^1,u_p^2,v_q^2,\ldots , u_p^t,v_q^t$ , giving each side of the boundary a forward-directed arrow. We denote this pointed oriented $2t$ -gon by $A_e = [ u_p^1,v_q^1,\ldots , u_p^t,v_q^t ]$ . Then glue the $A_e$ together in the manner of Definition 2.1 in order to obtain a $2t$ -polyhedron $P(G)$ (Figure 1).

Figure 1 Construction of a $2t$ -polyhedron. Give each side of a sequence of solid $2t$ -gons a direction and a label from one of two sets U, V, and then glue together corresponding sides with respect to their direction.

Definition 2.3. In this paper, we mainly concern ourselves with $4$ -polyhedra, that is, those constructed by gluing together squares. We refer to $4$ -polyhedra as tile complexes. For a connected bipartite graph G, write $TC(G)$ for the tile complex $P(G)$ , and define the set

(2-1) $$ \begin{align} \mathcal{S}(G):= \lbrace A_e = \,&[ u_p^1, v_q^1, u_p^2, v_q^2 ], [ \bar{u}_p^1, \bar{v}_q^2, \bar{u}_p^2, \bar{v}_q^1 ], \nonumber\\ &[ u_p^2,v_q^2,u_p^1,v_q^1 ], [ \bar{u}_p^2, \bar{v}_q^1, \bar{u}_p^1, \bar{v}_q^2 ] \mid e = u_p v_q \in E(G) \rbrace. \end{align} $$

We call the elements of $\mathcal {S}(G)$ pointed tiles. We define an equivalence relation which, for each $A_e$ , identifies the four corresponding pointed tiles in (2-1). We denote by $\mathcal {S}'(G)$ the quotient of $\mathcal {S}(G)$ with respect to this relation, and we write $A_e' = ( u_p^1,v_q^1,u_p^2,v_q^2 )$ for the equivalence class of $A_e$ in $\mathcal {S}'(G)$ . Then $\mathcal {S}'(G)$ is the set of geometric squares (that is, disregarding basepoint and orientation) in $TC(G)$ . We call elements of $\mathcal {S}'(G)$ unpointed tiles.

Notice that by placing the basepoint at the bottom-left vertex, we can arrange that the horizontal sides of each pointed tile are labelled by elements of U, and the vertical sides by elements of V, such that $\mathcal {S}(G) \subseteq U \times V \times U \times V$ . Indeed, the four tuples in (2-1) correspond to the four symmetries of a pointed tile that preserve this property (Figure 2).

Note also that, by design, any two pointed tiles in $\mathcal {S}(G)$ are distinct, and any two adjacent sides of a tile uniquely determine the remaining two sides.

Figure 2 Visualisation of tiles: $A=[x_1,y_1,x_2,y_2]$ , $B=[\bar {x}_1,\bar {y}_2,\bar {x}_2,\bar {y}_1]$ , and so on. These four pointed squares represent different pointed tiles, but the same unpointed tile.

Definition 2.4. Let G be a connected bipartite graph on $\alpha $ white and $\beta $ black vertices. Let U, V be sets with $|U| = 4\alpha $ , $|V| = 4\beta $ , as constructed above, and let $\mathcal {S} = \mathcal {S}(G) \subseteq U \times V \times U \times V$ be the corresponding set of pointed tiles. We call the datum $(G, U, V, \mathcal {S})$ a tile system.

This construction is closely related to, and indeed modelled on, that of a VH-datum, introduced in [Reference Wise16] and developed further in [Reference Burger and Mozes1].

3 The $C^{\star }$ -algebra corresponding to a tile system

Definition 3.1. Let $(G, U, V, \mathcal {S})$ be a tile system, and $A=[x_1,y_1,x_2,y_2]$ and $B=[x_3,y_3,x_4,y_4]$ be pointed tiles in $\mathcal {S}$ . We define two $4\alpha \beta \times 4\alpha \beta $ matrices $M_1,M_2$ with $AB$  th entry $M_i(A,B)$ as follows:

$$ \begin{align*} \begin{array}{r} M_1(A,B) = \left\{ \begin{array}{@{}l} 1 \quad \text{if}\ y_1 = \bar{y}_4\ \text{and}\ x_1 \neq \bar{x}_3, \\ 0 \quad \text{otherwise,} \end{array} \right.\\ \text{}\\ M_2(A,B) = \left\{ \begin{array}{@{}l} 1 \quad \text{if}\ x_2 = \bar{x}_3\ \text{and}\ y_1 \neq \bar{y}_3, \\ 0 \quad \text{otherwise,} \end{array} \right. \end{array} \end{align*} $$

as demonstrated in Figure 3. We call $M_1$ the horizontal adjacency matrix and $M_2$ the vertical adjacency matrix. If $M_i(A,B) = 1$ , we say that B is horizontally or vertically adjacent to A.

Figure 3 Horizontal and vertical adjacency: (a) $M_1(A,B) = 1$ , (b) $M_2(A,B) = 1$ .

Definition 3.2. Let $(G, U, V, \mathcal {S})$ be a tile system, and let A, B, C be pointed tiles in $\mathcal {S}(G)$ such that $M_1(A,B) = 1$ and $M_2(A,C) = 1$ . We say that the tile system $(G, U, V, \mathcal {S})$ satisfies the unique common extension property (UCE property) if there exists a unique $D \in \mathcal {S}$ such that $M_2(B,D) = M_1(C,D) = 1$ .

Proposition 3.3. Consider the complete bipartite graph $\kappa = \kappa (\alpha ,\beta )$ on $\alpha \geq 2$ white and $\beta \geq 2$ black vertices, and let $(\kappa , U, V, \mathcal {S}(\kappa ))$ be a tile system with corresponding adjacency matrices $M_1$ , $M_2$ . Then:

  1. (1) $M_1$ and $M_2$ are symmetric and commute with each other;

  2. (2) each row and column of $M_1$ and $M_2$ contains at least one nonzero element;

  3. (3) $(\kappa , U, V, \mathcal {S}(\kappa ))$ satisfies the UCE property.

Proof. It is straightforward to verify that the matrices $M_1$ and $M_2$ are symmetric. Now, consider the pointed tile $A = [ u_i^1,v_j^1,u_i^2,v_j^2 ] \in \mathcal {S}(\kappa )$ (any other tiles may be dealt with in a similar manner), and let $D = [ u_p^2,v_q^2,u_p^1,v_q^1 ] \in \mathcal {S}(\kappa )$ for some $p \neq i$ , $q \neq j$ . By the completeness of $\kappa $ and the fact that $\alpha , \beta \geq 2$ , there are tiles $B = [ \bar {u}_p^1, \bar {v}_j^2, \bar {u}_p^2, \bar {v}_j^1 ]$ and $C = [ \bar {u}_i^2, \bar {v}_q^1, \bar {u}_i^1, \bar {v}_q^2 ]$ in $\mathcal {S}(\kappa )$ such that $M_1(A,B) = M_2(B,D) = 1$ and $M_2(A,C) = M_1(C,D) = 1$ (Figure 4), proving (2). Since any two adjacent sides of a tile determine the remaining sides, B and C are unique. So $M_1 M_2 (A,D) = M_2 M_1 (A,D) \in \lbrace 0,1 \rbrace $ for all $A,D \in \mathcal {S}(\kappa )$ .

Figure 4 Proposition 3.3. Given tiles A and D as shown, tiles B and C are uniquely determined; hence, $M_1 M_2 = M_2 M_1$ . Likewise, given an initial tile A, a horizontally adjacent tile B and a vertically adjacent tile C, there is a unique tile D adjacent to both B and C: this is the UCE property.

Similarly, given $A,B,C \in \mathcal {S}(\kappa )$ as above, D is the unique tile adjacent to both B and C, and so $(\kappa , U, V, \mathcal {S}(\kappa ))$ has the UCE property.

We see shortly that a tile system is actually an example of a so-called k-rank graph (specifically a $2$ -rank graph), as introduced in [Reference Kumjian and Pask6] to build on work by [Reference Robertson and Steger10].

3.1 Higher-rank graphs

Definition 3.4. Let $\Lambda $ be a category such that $\operatorname {\mathrm {Ob}}(\Lambda )$ and $\operatorname {\mathrm {Hom}}(\Lambda )$ are countable sets (that is, a countable small category), and identify $\operatorname {\mathrm {Ob}}(\Lambda )$ with the identity morphisms in $\operatorname {\mathrm {Hom}}(\Lambda )$ . For a morphism $\lambda \in \operatorname {\mathrm {Hom}}_\Lambda (u,v)$ , we define range and source maps $r(\lambda ) = v$ and $s(\lambda ) = u$ , respectively.

Let $d : \Lambda \rightarrow \mathbb {N}^k$ be a functor, called the degree map, and let $\lambda \in \operatorname {\mathrm {Hom}}(\Lambda )$ . We call the pair $(\Lambda , d)$ a k-rank graph (or simply a k-graph) if, whenever $d(\lambda ) = \mathbf {m} + \mathbf {n}$ for some $\mathbf {m},\mathbf {n} \in \mathbb {N}^k$ , we can find unique elements $\mu , \nu \in \operatorname {\mathrm {Hom}}(\Lambda )$ such that $\lambda = \nu \mu $ , and $d({\kern1.3pt}\mu )=\mathbf {m}$ , $d(\nu ) = \mathbf {n}$ . Note that, for $\mu $ , $\nu $ to be composable, we must have $r({\kern1.3pt}\mu ) = s(\nu )$ .

For $\mathbf {n} \in \mathbb {N}^k$ , we write $\Lambda ^{\mathbf {n}} := d{}^{-1} (\mathbf {n})$ ; by the above property, $\Lambda ^{\mathbf {0}} = \operatorname {\mathrm {Ob}}(\Lambda )$ , and we call the elements of $\Lambda ^{\mathbf {0}}$ the vertices of $(\Lambda , d)$ ; see [Reference Kumjian and Pask6].

We direct the reader to [Reference Sims, Raeburn and Yeend14], for example, for further details and standard examples of higher-rank graphs.

If E is a directed graph on n vertices, we can construct an $n \times n$ vertex matrix $M_E(i,j)$ with $ij$  th entry $1$ if there is an edge from i to j, and $0$ otherwise.

If E, F are directed graphs with the same vertex set, whose associated vertex matrices $M_E$ , $M_F$ commute, then [Reference Kumjian and Pask6] showed that we can construct a $2$ -rank graph out of E and F. We use their method to prove the following proposition.

Proposition 3.5. Let $\kappa = \kappa (\alpha ,\beta )$ be the complete bipartite graph on $\alpha \geq 2$ white and $\beta \geq 2$ black vertices, and let $(\kappa , U, V, \mathcal {S}(\kappa ))$ be a tile system with adjacency matrices $M_1$ , $M_2$ . Then $(\kappa, U,V, \mathcal{S}(\kappa))$ has a 2-rank graph structure.

Proof. Following the method of Theorem 2.2, label the elements of the sets U, V

$$ \begin{align*} U &= \lbrace u_1^1,u_1^2,\ldots , u_\alpha^1,u_\alpha^2, \bar{u}_1^1,\bar{u}_1^2,\ldots , \bar{u}_\alpha^1,\bar{u}_\alpha^2 \rbrace, \\ \quad V &= \lbrace v_1^1,v_1^2,\ldots , v_\beta^1,v_\beta^2, \bar{v}_1^1,\bar{v}_1^2, \ldots , \bar{v}_\beta^1,\bar{v}_\beta^2 \rbrace, \end{align*} $$

where $u_1,\ldots , u_\alpha $ and $v_1,\ldots ,v_\beta $ are the white and black vertices of $\kappa $ , respectively. Construct the tile complex $TC(\kappa )$ , and consider the set $\mathcal {S}(\kappa ) \subseteq U \times V \times U \times V$ of pointed tiles of $TC(\kappa )$ . Since $\kappa $ is complete, there is for an edge joining each $u_i$ and $v_j$ , thus:

$$ \begin{align*} \mathcal{S}(\kappa) = \lbrace &[ u_i^1,v_j^1,u_i^2,v_j^2 ], [ \bar{u}_i^1, \bar{v}_j^2, \bar{u}_i^2, \bar{v}_j^1 ], \\ &[ u_i^2,v_j^2,u_i^1,v_j^1 ], [ \bar{u}_i^2, \bar{v}_j^1, \bar{u}_i^1, \bar{v}_j^2 ] \mid\, 1 \leq i \leq \alpha, 1\leq j \leq \beta \rbrace. \end{align*} $$

Consider the corresponding adjacency matrices $M_1$ and $M_2$ as described in Definition 3.1, and note that they commute by Proposition 3.3. We can draw directed graphs E, F with the same vertex set $E^0 = F^0 = \mathcal {S}(\kappa )$ . We draw a directed edge in E from A to B whenever $M_1(A,B) = 1$ , and in F from A to B whenever $M_2(A,B) = 1$ (Figure 5). Write $r_E$ , $s_E$ (respectively, $r_F$ , $s_F$ ) for the maps describing the respective range and source of edges in $E^1$ (respectively, $F^1$ ).

Figure 5 Visualisation of the tile system corresponding to the complete bipartite graph $\kappa (2,2)$ . Each vertex is labelled with an element of $\mathcal {S}(\kappa )$ ; a few labels have been shown here. A solid (dashed) arrow joins vertex A to B if and only if $M_1(A,B)=1$ ( $M_2(A,B)=1$ , respectively). Notice the commuting squares, giving the tile system a $2$ -rank graph structure: from any vertex A, follow a solid arrow, and then a dashed arrow to another vertex D, say. Then $\theta $ defines a unique dashed-solid path from A to D. The $1$ -skeleton of the $2$ -rank graph $\Lambda (\kappa (\alpha ,\beta ))$ is strongly connected only when $\alpha , \beta \geq 3$ .

Define the following edge sets: $E^1 \ast F^1 := \lbrace (\lambda ,\mu ) \in E^1 \times F^1 \mid r_E(\lambda ) = s_F({\kern1.3pt}\mu ) \rbrace $ and $F^1 \ast E^1 := \lbrace ({\kern1.3pt}\mu ,\lambda ) \in F^1 \times E^1 \mid r_F({\kern1.3pt}\mu ) = s_E(\lambda ) \rbrace $ . Since $M_1$ , $M_2$ commute, there is a unique bijection $\theta : E^1 \ast F^1 \rightarrow F^1 \ast E^1$ , mapping $(\lambda ,\mu ) \mapsto ({\kern1.3pt}\mu ',\lambda ')$ such that ${s_E(\lambda ) = s_F({\kern1.3pt}\mu ')}$ and $r_F({\kern1.3pt}\mu ) = r_E(\lambda ')$ .

We construct a $2$ -rank graph $(\Lambda ,d)$ in the following way. Let $\Lambda ^0 = \mathcal {S}(\kappa )$ , and for each $(m,n) \in \mathbb {N}^2$ , define the set $W(m,n) := \lbrace (p,q) \in \mathbb {N}^2 \mid p \leq m, q \leq n \rbrace $ . Then an element of $\Lambda ^{(m,n)}$ is a triple $(A,\lambda ,\mu ) = ((A(p,q))_{p,q},(\lambda (p,q))_{p,q},({\kern1.3pt}\mu (p,q))_{p,q})$ such that:

  1. (a) $A(p,q) \in \mathcal {S}(\kappa )$ for some $(p,q) \in W(m,n)$ ;

  2. (b) $\lambda (p,q) \in E^1$ for some $(p,q) \in W(m-1,n)$ ;

  3. (c) $\mu (p,q) \in F^1$ for some $(p,q) \in W(m,n-1)$ ;

  4. (d) $s_E(\lambda (p,q)) = s_F({\kern1.3pt}\mu (p,q)) = A(p,q)$ ;

  5. (e) $r_E(\lambda (p,q)) = A(p+1,q)$ and $r_F({\kern1.3pt}\mu (p,q)) = A(p,q+1)$ ;

  6. (f) $\theta (\lambda (p,q),\mu (p+1,q)) = ({\kern1.3pt}\mu (p,q),\lambda (p,q+1))$ ,

whenever these conditions make sense. We write $\Lambda := \bigcup _{m,n \geq 0} \Lambda ^{(m,n)}$ , and define range and source maps $r(A,\lambda ,\mu ) := A(0,0)$ , $s(A,\lambda ,\mu ):= A(m,n)$ , respectively. Note that two finite paths $\mu $ , $\nu $ in such a directed graph E can be concatenated to give a path $\nu \cdot \mu $ if and only if $s_E({\kern1.3pt}\mu ) = r_E(\nu )$ ; consequently, ‘change the direction’ of the sources and ranges of the arrows here.

If $\varphi $ , $\psi $ are paths of nonzero length m, n in E, F respectively, with $r_E(\varphi ) = s_F(\psi )$ , then there is a unique element $\varphi \psi = (A,\lambda ,\mu ) \in \Lambda ^{(m,n)}$ such that $\varphi = \lambda (0,0) \cdots \lambda (m-1,0)$ and $\psi = \mu (m,0) \cdots \mu (m,n-1)$ . If, instead (or as well), $r_F(\psi ) = s_E(\varphi )$ , then there is a unique element $\psi \varphi $ such that $\varphi = \lambda (0,n) \cdots \lambda (m-1,n)$ and $\psi = \mu (0,0) \cdots \mu (0,n-1)$ (Figure 6).

Figure 6 An element $(A,\lambda ,\mu )$ of $\Lambda ^{(m,n)}$ can be represented as an $m \times n$ grid. The isomorphism $\theta $ defines commuting squares. Here is an element of $\Lambda ^{(2,5)}$ .

Then, given two elements $( A_1,\lambda _1,\mu _1) \in \Lambda ^{( m_1,n_1)}$ and $( A_2,\lambda _2, \mu _2) \in \Lambda ^{( m_2,n_2)}$ such that $A_1 ( m_1,n_1) = A_2(0,0)$ , we can find a unique element $( A_1,\lambda _1,\mu _1)( A_2,\lambda _2,\mu _2) = ( A_3,\lambda _3,\mu _3 )$ in $\Lambda ^{( m_1+m_2,n_1+n_2)}$ that satisfies

  1. (a) $A_3(p,q) = A_1(p,q)$ , and $A_3(m+p,n+q) = A_2(p,q)$ ;

  2. (b) $\lambda _3(p,q) = \lambda _1(p,q)$ , and $\lambda _3(m+p,n+q) = \lambda _2(p,q)$ ;

  3. (c) $\mu _3(p,q) = \mu _1(p,q)$ , and $\mu _3(m+p,n+q) = \mu _2(p,q)$ ,

whenever these conditions make sense. In this way, composition is defined in $\Lambda $ , and, by construction, we have associativity and the factorisation property of Definition 3.4. Thus $\Lambda $ , together with obvious degree functor $d : (A,\lambda ,\mu ) \mapsto (m,n)$ for $(A, \lambda , \mu ) \in \Lambda ^{(m,n)}$ , has the structure of a $2$ -rank graph, and we write $(\Lambda ,d) = \Lambda (\kappa )$ .

Definition 3.6. Let $(\Lambda ,d)$ be a k-rank graph, let $\mathbf {n} \in \mathbb {N}^k$ , and let $v \in \Lambda ^{\mathbf {0}}$ . Write $\Lambda ^{\mathbf {n}}(v)$ for the set of morphisms in $\Lambda ^{\mathbf {n}}$ which map onto v, that is, $\Lambda ^{\mathbf {n}}(v) := \lbrace \lambda \in \Lambda ^{\mathbf {n}} \mid r(\lambda ) = v\rbrace $ . We say that $(\Lambda ,d)$ is row-finite if each set $\Lambda ^{\mathbf {n}}(v)$ is finite, and that $(\Lambda ,d)$ has no sources if each $\Lambda ^{\mathbf {n}}(v)$ is nonempty.

As an extension of the concept of a graph algebra (see [Reference Raeburn9]), we can associate a $C^{\star }$ -algebra to a k-rank graph.

Definition 3.7. Let $\Lambda = (\Lambda ,d)$ be a row-finite k-rank graph with no sources. We define $C^{\star }(\Lambda )$ to be the universal $C^{\star }$ -algebra generated by a family $\lbrace s_\lambda \mid \lambda \in \Lambda \rbrace $ of partial isometries satisfying the following properties.

  1. (a) For all $u,v \in \Lambda^\mathbf{0}$ , we have $(s_v)^2 = s_v = s_v^*$ and $s_us_v = 0$ whenever $u \neq v$ .

  2. (b) If $r(\lambda ) = s({\kern1.3pt}\mu )$ for some $\lambda , \mu \in \Lambda $ , then $s_{\mu \lambda } = s_\mu s_\lambda $ .

  3. (c) For all $\lambda \in \Lambda $ , we have $s_\lambda ^* s_\lambda = s_{s(\lambda )}$ .

  4. (d) For all vertices $v \in \Lambda ^{\mathbf {0}}$ and $\mathbf {n} \in \mathbb {N}^k$ ,

    $$ \begin{align*} s_v = \sum_{\lambda \in \Lambda^{\mathbf{n}}(v)} s_\lambda s_\lambda^*. \end{align*} $$

Note that, without the row-finiteness condition, property (d) is not well defined.

Theorem 3.8 (Evans [Reference Evans2, Proposition 4.4])

Let $\Lambda $ be a row-finite $2$ -graph with no sources, finite vertex set $\Lambda ^{\mathbf {0}}$ , and vertex matrices $M_E$ , $M_F$ . Then

$$ \begin{align*} K_0 (C^{\star} (\Lambda)) &\cong \mathbb{Z}^{r_0} \oplus \operatorname{\mathrm{tor}}(\operatorname{\mathrm{coker}}(\mathbf{1}-M_E^T, \mathbf{1}-M_F^T)), \\ K_1(C^{\star} (\Lambda)) &\cong \mathbb{Z}^{r_1} \oplus \operatorname{\mathrm{tor}}(\operatorname{\mathrm{coker}}(\mathbf{1}-M_E, \mathbf{1}-M_F)), \end{align*} $$


$$ \begin{align*} r_0 &:= \operatorname{\mathrm{rk}}(\operatorname{\mathrm{coker}}(\mathbf{1}-M_E^T, \mathbf{1}-M_F^T)) + \operatorname{\mathrm{rk}}(\operatorname{\mathrm{coker}} (\mathbf{1}-M_E,\mathbf{1}-M_F) ), \\ r_1 &:= \operatorname{\mathrm{rk}}(\operatorname{\mathrm{coker}}(\mathbf{1}-M_E^T, \mathbf{1}-M_F^T)) + \operatorname{\mathrm{rk}}(\operatorname{\mathrm{coker}}(\mathbf{1}-M_E,\mathbf{1}-M_F) ), \end{align*} $$

$| \Lambda ^{\mathbf {0}}| = n$ , $\mathbf {1}$ is the $n \times n$ identity matrix, $({}\ast {} ,{} \ast {})$ denotes the corresponding block $n \times 2n$ matrix, $\operatorname {\mathrm {rk}}(\mathfrak {G})$ is the torsion-free rank of a finitely generated Abelian group $\mathfrak {G}$ , and $\operatorname {\mathrm {tor}}(\mathfrak {G})$ is the torsion part of $\mathfrak {G}$ .

Corollary 3.9. Let $\kappa = \kappa (\alpha ,\beta )$ be the complete bipartite graph on $\alpha \geq 2$ white and $\beta \geq 2$ black vertices, and let $(\kappa ,U,V,\mathcal {S}(\kappa ))$ be a tile system with adjacency matrices $M_1$ , $M_2$ as in Definition 3.1. For simplicity, we write $C^{\star }(\kappa ) = C^{\star }(\Lambda (\kappa ))$ . Then

$$ \begin{align*} K_0 (C^{\star} (\kappa)) = K_1 (C^{\star} (\kappa)) = \operatorname{\mathrm{coker}}(\mathbf{1}-M_1^T, \mathbf{1}-M_2^T) \oplus \mathbb{Z}^r, \end{align*} $$

where $r := \operatorname {\mathrm {rk}} (\operatorname {\mathrm {coker}}(\mathbf {1}-M_1^T, \mathbf {1}-M_2^T))$ .

Proof. Firstly, $\alpha , \beta < \infty $ by assumption, and by the UCE property of the tile system (Proposition 3.3) we know that each row and column of $M_1$ and $M_2$ has at least one nonzero element. Hence, $\Lambda (\kappa )$ is row-finite, has no sources, and satisfies $|\Lambda (\kappa )^{\mathbf {0}}| = 4\alpha \beta $ , whence the result follows from Theorem 3.8.

Theorem 3.10 (K-groups for pointed tile systems)

Let $a,b \ge 0$ . Let ${\kappa (a+2,b+2)}$ be the complete bipartite graph on $a+2$ white and $b+2$ black vertices. Without loss of generality, we assume that $a\leq b$ . Write $l := \operatorname {\mathrm {lcm}} ( a, b )$ , and $g := \gcd ( a, b )$ . Then, for $\epsilon = 0,1$ , we have the following assertions.

  1. (1) If $a=b=0$ , then $K_\epsilon (C^{\star } (\kappa (a+2,b+2))) = K_\epsilon (C^{\star } (\kappa (2,2))) \cong \mathbb {Z}^8$ .

  2. (2) If $a=0,1$ and $b \geq 1$ , then

    $$ \begin{align*} K_\epsilon (C^{\star} (\kappa (a+2,b+2))) \cong (\mathbb{Z}/b)^2 \oplus \mathbb{Z}^{4(b+1)}. \end{align*} $$
  3. (3) If $a,b \geq 2$ and $a,b$ are coprime, then

    $$ \begin{align*} K_\epsilon (C^{\star} (\kappa (a+2,b+2))) \cong (\mathbb{Z}/a)^{b-a} \oplus ( \mathbb{Z}/ab )^{a+1} \oplus \mathbb{Z}^{2(a+1)(b+1)}. \end{align*} $$
  4. (4) If $a,b \geq 2$ and $a,b$ are not coprime, then

    $$ \begin{align*} K_\epsilon(C^{\star} (\kappa(a+2,b+2))) \cong (\mathbb{Z}/a)^{b-a} \oplus ( \mathbb{Z}/l )^{a+1} \oplus ( \mathbb{Z}/g)^{a+2} \oplus \mathbb{Z}^{2(a+1)(b+1)}, \end{align*} $$
    where $(\mathbb {Z}/a)^0$ is defined to be the trivial group in the case that $a=b$ .

Proof. We begin by proving (3) and (4), since (1) and (2) are special cases thereof.

Assume that $a,b \geq 2$ . Write $\alpha = a+2$ and $\beta = b+2$ , and for $1 \leq i \leq \alpha $ , $1 \leq j \leq \beta $ , let $A_{ij}$ denote the pointed tile $[ u_i^1,v_j^1,u_i^2,v_j^2] \in \mathcal {S}(\kappa )$ . Similarly, write $B_{ij} := [ \bar {u}_i^1, \bar {v}_j^2, \bar {u}_i^2, \bar {v}_j^1 ]$ , $C_{ij} := [ \bar {u}_i^2, \bar {v}_j^1, \bar {u}_i^1, \bar {v}_j^2 ]$ , $D_{ij} := [ u_i^2,v_j^2,u_i^1,v_j^1 ]$ for the tiles with the same edge labels as the horizontal reflection, vertical reflection, and rotation by $\pi $ of $A_{ij}$ , respectively. Then $\mathcal {S}(\kappa ) = \lbrace A_{ij},B_{ij},C_{ij},D_{ij} \mid 1 \leq i \leq \alpha , 1 \leq j \leq \beta \rbrace $ , and

(3-1) $$ \begin{align} \operatorname{\mathrm{coker}} = \bigg\langle S \in \mathcal{S}(\kappa) \bigg\vert\, S = \sum_{T \in \mathcal{S}(\kappa)} M_1(S,T)\cdot T = \sum_{T \in \mathcal{S}(\kappa)} M_2(S,T)\cdot T \bigg\rangle , \end{align} $$

where $\operatorname {\mathrm {coker}} := \operatorname {\mathrm {coker}}(\mathbf {1}-M_1^T,\mathbf {1}-M_2^T)$ . Now fix some $p \in \lbrace 1 ,\ldots , \alpha \rbrace $ and $q \in \lbrace 1 , \ldots , \beta \rbrace $ , and notice that:

  • $M_1(A_{pq}, T) = 1$ if and only if $T = B_{iq}$ , and $M_1(B_{pq}, T) = 1$ if and only if $T = A_{iq}$ , for some $i \neq p$ ;

  • $M_1(C_{pq}, T) = 1$ if and only if $T = D_{iq}$ , and $M_1(D_{pq}, T) = 1$ if and only if $T = C_{iq}$ , for some $i \neq p$ ;

  • $M_2(A_{pq}, T) = 1$ if and only if $T = C_{pj}$ , and $M_2(B_{pq}, T) = 1$ if and only if $T = D_{pj}$ , for some $j \neq q$ ;

  • $M_2(C_{pq}, T) = 1$ if and only if $T = A_{pj}$ , and $M_2(D_{pq}, T) = 1$ if and only if $T = B_{pj}$ , for some $j \neq q$ .

Hence, the relations of (3-1) are equations of the form $A_{pq} = \sum _{i \neq p} B_{iq} = \sum _{j \neq q} C_{pj}$ , and so on for each $B_{pq}$ , $C_{pq}$ , $D_{pq}$ . In particular, we can write $B_{pq} = \sum _{i \neq p} A_{iq}$ and $C_{pq} = \sum _{j \neq q} A_{pj}$ so that

$$ \begin{align*} A_{pq} = (\alpha - 1)A_{pq} + (\alpha - 2)\sum_{i \neq p}A_{iq} \quad\text{and}\quad A_{pq} = (\beta - 1)A_{pq} + (\beta - 2)\sum_{j \neq q}A_{pj}. \end{align*} $$

Define $J_q := \sum _{i=1}^\alpha A_{iq}$ , and $I_p := \sum _{j=1}^\beta A_{pj}$ . Then $(\alpha - 2)J_q = (\beta - 2)I_p = 0$ , and, viewing the sum of all the tiles $A_{ij}$ both as the sum of all the $I_i$ and of the $J_j$ , we conclude that $g\Sigma = 0$ , where $\Sigma := \sum _{i,j} A_{ij}$ .

Now, we can also write $D_{pq}$ (and all of the relevant relations) in terms of the $A_{ij}$ , namely $D_{pq} = \sum _{i \neq p} \sum _{j \neq q} A_{ij}$ . Hence, we can remove all the $B_{pq}$ , $C_{pq}$ , and $D_{pq}$ from the list of generators of $\operatorname {\mathrm {coker}}$ , yielding

(3-2) $$ \begin{align} \operatorname{\mathrm{coker}} &= \langle A_{pq} \mid\, (\alpha -2)J_q = (\beta -2)I_p = 0, J_q = \textstyle\sum_i A_{iq}, \nonumber\\ & \quad I_p = \textstyle\sum_j A_{pj}, \text{ for } 1 \leq p \leq \alpha, 1 \leq q \leq \beta \rangle. \end{align} $$

We have the following equalities:

$$ \begin{align*} A_{p1} = I_p - \sum_{j=2}^\beta A_{pj},\quad A_{1q} = J_q - \sum_{i=2}^\alpha A_{iq},\quad I_1 = \Sigma - \sum_{i=2}^\alpha I_i,\quad J_1 = \Sigma - \sum_{j=2}^\beta J_j. \end{align*} $$

Furthermore, $A_{11}$ may be expressed in terms of $\Sigma $ , $I_p$ , $J_q$ , and $A_{pq}$ for $p,q \geq 2$ , and so, after a sequence of Tietze transformations on (3-2), we find that

$$ \begin{align*} \operatorname{\mathrm{coker}} = \langle \Sigma, I_p, J_q, A_{pq} \mid (\alpha -2)J_q = (\beta -2)I_p = g\Sigma = 0, \text{ for } 2 \leq p \leq \alpha, 2 \leq q \leq \beta\rangle , \end{align*} $$

where $g:= \gcd (\alpha -2,\beta -2)$ . This, after substituting $a = \alpha -2$ and $b = \beta -2$ , gives a presentation for $(\mathbb {Z}/b )^{a+1} \oplus ( \mathbb {Z}/a )^{b+1} \oplus ( \mathbb {Z}/g) \oplus \mathbb {Z}^{(a+1)(b+1)}$ . In particular, we have $a+1$ copies of $( \mathbb {Z}/b ) \oplus ( \mathbb {Z}/a )$ . It is well known that if a and b are not coprime, $( \mathbb {Z}/b ) \oplus ( \mathbb {Z}/a ) \cong ( \mathbb {Z}/l ) \oplus ( \mathbb {Z}/g )$ ; in case (4), this, together with Corollary 3.9, immediately gives the desired result. In case (3), where a and b are coprime, we instead have that $( \mathbb {Z}/b ) \oplus ( \mathbb {Z}/a ) \cong ( \mathbb {Z}/ab)$ , and we are done.

Now consider case (1), where $\alpha = \beta = 2$ . Then, following the method above, $\operatorname {\mathrm {coker}}$ is generated by $\lbrace A_{pq} \mid p,q = 1,2 \rbrace $ with trivial relations, and so $\operatorname {\mathrm {coker}} \cong \mathbb {Z}^4$ . Hence, by Corollary 3.9, $K_\epsilon (C^{\star }(\kappa )) \cong \mathbb {Z}^8$ .

Similarly, when $\alpha = 2$ and $\beta \geq 3$ , it is straightforward to show that

$$ \begin{align*} \operatorname{\mathrm{coker}} = \langle I_p, A_{pq} \mid (\beta - 2)I_p = 0, \text{ for } p =1,2 \text{ and } 2 \leq q \leq \beta\rangle, \end{align*} $$

and, when $\alpha = 3$ and $\beta \geq 3$ ,

$$ \begin{align*} \operatorname{\mathrm{coker}} = \langle \Sigma,I_p,J_q, A_{pq} \mid J_q = (\beta - 2)I_p = \Sigma = 0, \text{ for } p =2,3 \text{ and } 2 \leq q \leq \beta\rangle, \end{align*} $$

both of which are presentations of $(\mathbb {Z}/(\beta - 2))^2 \oplus \mathbb {Z}^{2(\beta - 1)}$ ; hence, by Corollary 3.9, (2) is proved.

Example 3.11. Recall the tile system corresponding to $\kappa (2,2)$ , given in Figure 5. From the diagram, we can see that the ( $1$ -skeleton of the) $2$ -rank graph $\Lambda (\kappa (2,2))$ comprises four connected components, each being the Cartesian product $C_2 \times C_2$ , depicted in Figure 7. It is well known that the k-graph $C^{\star }$ -algebra of $C_2$ is isomorphic to $M_2(C(\mathbb {T}))$ . Furthermore, there is a natural isomorphism $C^{\star }(C_m \times C_n) \cong M_{mn}(C(\mathbb {T}^2))$ , and so $C^{\star }(\kappa (2,2)) \cong (M_4(C(\mathbb {T}^2)))^4$ . The K-groups of this $C^{\star }$ -algebra are both $\mathbb {Z}^8$ , in agreement with Theorem 3.10.

Figure 7 The $2$ -graph $\Lambda (\kappa (2,2))$ , depicted in Figure 5, consists of four copies of $C_2 \times C_2$ , where $C_2$ is the cyclic $1$ -graph with two vertices.

Theorem 3.12. Let $\alpha , \beta \geq 3$ , and let $\kappa = \kappa (\alpha , \beta )$ be the complete bipartite graph on $\alpha $ white and $\beta $ black vertices. Then the order of the class of the identity $[\mathbf {1}]$ in $K_0(C^{\star }(\Lambda (\kappa )))$ is equal to $g := \gcd (\alpha - 2, \beta - 2)$ .

Proof. From [Reference Kimberley and Robertson3], it follows that the order of $[\mathbf {1}]$ in $K_0(C^{\star }(\kappa ))$ is equal to the order of the sum of pointed tiles in $\mathcal {S}(\kappa )$ ; by considerations in the proof of Theorem 3.10, we know this to be g.

4 Aperiodicity and Kirchberg–Phillips classification

Kumjian and Pask in [Reference Kumjian and Pask6] developed conditions under which the $C^{\star }$ -algebra of a k-rank graph is both simple and purely infinite. In this section we show that the conditions are satisfied by the algebras $C^{\star }(\kappa )$ , and thus, by the Kirchberg and Phillips results [Reference Kirchberg4, Reference Phillips8], that the $C^{\star }(\kappa )$ are completely classified by their K-theory. We detail the following definitions from [Reference Kumjian and Pask6].

Let $k \geq 1$ , and let $\Omega _k$ be the countable small category with object set $\operatorname {\mathrm {Ob}}(\Omega _k) := \mathbb {N}_0^k$ and morphism set $\operatorname {\mathrm {Hom}}(\Omega _k)$ given by

$$ \begin{align*} \lbrace (\mathbf{m}, \mathbf{n}) = (m_1, \ldots , m_k, n_1,\ldots , n_k) \in \mathbb{N}_0^k \times \mathbb{N}_0^k\, \vert\, m_i \leq n_i \text{ for all } 1 \leq i \leq k \rbrace. \end{align*} $$

We identify $\operatorname {\mathrm {Ob}}(\Omega _k)$ with the set of identity morphisms $\lbrace (\mathbf {m}, \mathbf {m}) \mid \mathbf {m} \in \mathbb {N}_0^k \rbrace $ , and, hence, identify $\Omega _k$ with $\operatorname {\mathrm {Hom}}(\Omega _k)$ . Define range and source maps $r(\mathbf {m},\mathbf {n}) := \mathbf {m}$ and $s(\mathbf {m},\mathbf {n}) := \mathbf {n}$ , respectively. Then $\Omega _k$ , together with the degree map $d(\mathbf {m},\mathbf {n}) := \mathbf {n} - \mathbf {m}$ , is a k-rank graph, which we can visualise as a nonnegative integer lattice in $\mathbb {R}^k$ (see Figure 6).

Definition 4.1. Let $\Lambda $ be a k-rank graph. We define the infinite path space $\Lambda ^\infty $ of $\Lambda $ to be $\Lambda ^\infty := \lbrace \varphi : \Omega _k \rightarrow \Lambda \mid \varphi \text { is a}\ k\text {-graph morphism} \rbrace $ .

Given a vertex $v \in \Lambda ^{\mathbf {0}}$ , we write $\Lambda ^\infty (v)$ for the set of infinite paths that begin at v, that is, $\Lambda ^\infty (v) := \lbrace \varphi \in \Lambda ^\infty \mid \varphi (\mathbf {0}) = v \rbrace $ .

Let $\mathbf {p} \in \mathbb {Z}^k$ , and let $\varphi \in \Lambda ^\infty $ . We say that $\mathbf {p}$ is a period for $\varphi $ if, for every $(\mathbf {m},\mathbf {n}) \in \Omega _k$ with $\mathbf {m} + \mathbf {p} \geq \mathbf {0}$ , we have $\varphi (\mathbf {m}+\mathbf {p}, \mathbf {n}+\mathbf {p}) = \varphi (\mathbf {m}, \mathbf {n})$ . We call $\varphi $ periodic if we can find a nonzero period for $\varphi $ .

Given $\mathbf {q} \in \mathbb {N}_0^k$ and a path $\varphi \in \Lambda ^\infty $ , we write $\varphi _{\mathbf {q}}(\mathbf {m},\mathbf {n}) := (\mathbf {m}+\mathbf {q}, \mathbf {n}+\mathbf {q})$ . We say that $\varphi $ is eventually periodic if we can find some nonzero $\mathbf {q} \in \mathbb {N}_0^k$ such that $\varphi _{\mathbf {q}}$ is periodic. We say that an infinite path $\varphi $ is aperiodic if it is neither periodic nor eventually periodic.

We say that $\Lambda $ satisfies the aperiodicity condition (also referred to in the literature as Condition (A)) if, for every vertex $v \in \Lambda ^{\mathbf {0}}$ , we can find an aperiodic path $\varphi \in \Lambda ^\infty (v)$ . We say that $\Lambda $ is cofinal if, for every vertex $v \in \Lambda ^{\mathbf {0}}$ and every infinite path $\varphi \in \Lambda ^\infty $ , we can find $\lambda \in \Lambda $ and $\mathbf {n} \in \mathbb {N}_0^k$ such that $r(\lambda ) = v$ and $s(\lambda ) = \varphi (\mathbf {n})$ .

The aperiodicity condition is a generalisation of the condition on $1$ -graphs that every cycle has an entrance. Similarly, cofinality is a generalisation of the property of $1$ -graphs that every vertex be reachable from somewhere on every infinite path.

Lemma 4.2. Consider the complete bipartite graph $\kappa = \kappa (\alpha ,\beta )$ , where $\alpha ,\beta \geq 3$ , and let $\Lambda (\kappa )$ be the corresponding $2$ -rank graph as constructed in the proof of Proposition 3.5. Then $\Lambda (\kappa )$ satisfies the aperiodicity condition.

In order to get a feeling as to why this is true, consider Figure 8, which shows a representation of $\Lambda (\kappa (3,3))$ . Each vertex is labelled by a pointed tile from $\mathcal {S}(\kappa (3,3))$ , and since each tile is vertically adjacent to two others (and horizontally adjacent to two others), there are two solid arrows and two dashed arrows emanating from each vertex of $\Lambda (\kappa (3,3))$ . This suggests that, analogously to the $1$ -graph condition, we can always find an entrance to any cycle in $\Lambda $ , namely by stopping mid-cycle at a vertex, and diverting the path down the second of the two available edges. Hence, as long as $\alpha , \beta \geq 3$ , there is enough choice at each vertex to be able to exit a cycle.

Figure 8 A representation of $\Lambda (\kappa (3,3))$ . It is always possible to exit a cycle.

Proof. Firstly, write $\Lambda = \Lambda (\kappa )$ , and let $A \in \Lambda ^{\mathbf {0}}$ be an arbitrary vertex. We construct an aperiodic infinite path beginning from A in the following way.

Let $x:\Omega _1 \rightarrow \bigcup _{m \geq 0} \Lambda ^{(m,0)}$ be a $1$ -graph morphism such that $x(0) = A$ . The vertex A represents a pointed tile in $\mathcal {S}(\kappa )$ , which is horizontally adjacent to $\beta - 1$ other pointed tiles. Hence, A is connected by bidirectional blue arrows to $\beta - 1$ other vertices in $\Lambda $ . Choose two of these vertices, $B_1$ and $B_2$ , say, and let x be such that

$$ \begin{align*} x(m,m) = \begin{cases} A & \text{if}\ m\ \text{is even,} \\ B_1 & \text{if}\ m = r^2 + r + 1\ \text{for some}\ r \geq 1, \\ B_2 & \text{otherwise,} \end{cases} \end{align*} $$

for all $m \in \mathbb {N}_0$ . Since $x$ forms an aperiodic sequence, there is no $p \in \mathbb {Z}$ such that $x(m,m) = x(m+p,m+p)$ for all m, nor any $q \in \mathbb {N}$ such that $x_q$ is periodic; hence, x is an aperiodic path. Similarly, define $y : \Omega _1 \rightarrow \bigcup _{n \geq 0} \Lambda ^{(0,n)}$ by

$$ \begin{align*} y(n,n) = \begin{cases} A & \text{if}\ n\ \text{is even,} \\ C_1 & \text{if}\ n = s^2 + s + 1,\ \text{for some}\ s \geq 1, \\ C_2 & \text{otherwise,} \end{cases} \end{align*} $$

for some vertices labelled by pointed tiles $C_1$ , $C_2$ that are vertically adjacent to A. Then y is also an aperiodic path. By the UCE property, x and y uniquely determine an infinite path $\varphi : \Omega _2 \rightarrow \Lambda $ with $\varphi ((m,0),(m,0)) = x(m,m)$ and $\varphi ((0,n),(0,n)) = y(n,n)$ .

Denote by D the unique pointed tile (other than A) adjacent to both $B_1$ and $C_1$ . This cannot also be adjacent to $B_2$ , nor to $C_2$ , so $\varphi ((m,n),(m,n)) = D$ precisely when $m = r^2+r+1$ and $n=s^2+s+1$ , for some $r,s \geq 1$ . As above, there is no $\mathbf {p} \in \mathbb {Z}^2$ such that $\varphi ((m,n),(m,n)) = \varphi ((m,n)+\mathbf {p},(m,n)+\mathbf {p})$ , nor any $\mathbf {q} \in \mathbb {N}_0^2$ such that $\varphi _{\mathbf {q}}$ is periodic. Since our initial vertex A was arbitrary, we are done.

The following definitions are required for the rest of the section. For the reader who desires more detail, we recommend [Reference Rørdam, Larsen and Laustsen11, Ch. 5].

Definition 4.3. Let $\mathcal {A}$ be a unital $C^{\star }$ -algebra, and let $\mathcal {B} \subset \mathcal {A}$ be a $C^{\star }$ -subalgebra. We say that $\mathcal {B}$ is hereditary if, for all $a,b \in \mathcal {A}$ , if $b \in \mathcal {B}$ and $a \leq b$ , then $a \in \mathcal {B}$ .

We say that $\mathcal {A}$ is simple if it has no nontrivial closed two-sided ideals.

If $\mathcal {A}$ is simple, we say that it is purely infinite if every nonzero hereditary $C^{\star }$ -subalgebra of $\mathcal {A}$ contains a projection which is Murray–von Neumann equivalent to a proper subprojection of itself. Equivalently, $\mathcal {A}$ is purely infinite if every nonzero hereditary $C^{\star }$ -subalgebra contains a projection equivalent to $\mathbf {1}$ .

Theorem 4.4 (Kumjian and Pask [Reference Kumjian and Pask6])

Let $\Lambda $ be a k-rank graph that satisfies the aperiodicity condition. Then the associated universal $C^{\star }$ -algebra $C^{\star }(\Lambda )$ is simple if and only if $\Lambda $ is cofinal.

Theorem 4.5 (Kumjian and Pask [Reference Kumjian and Pask6], Sims [Reference Sims13, Proposition 8.8])

Let $\Lambda $ be a k-rank graph that is cofinal and satisfies the aperiodicity condition. Suppose that, for every $v \in \Lambda ^{\mathbf {0}}$ , we can find $\lambda \in \Lambda $ with $r(\lambda ) = v$ , and some cycle $\mu \in \Lambda $ with an entrance, such that $d({\kern1.3pt}\mu ) \neq \mathbf {0}$ , and $s(\lambda ) = r({\kern1.3pt}\mu ) = s({\kern1.3pt}\mu )$ . Then $C^{\star }(\Lambda )$ is purely infinite.

Proposition 4.6. Consider $\kappa = \kappa (\alpha ,\beta )$ for $\alpha ,\beta \geq 3$ , and let $\Lambda (\kappa )$ be the corresponding $2$ -rank graph. Then the $C^{\star }$ -algebra $C^{\star }(\kappa )$ from Definition 3.7 is simple and purely infinite.

Proof. Firstly observe that $\Lambda (\kappa )$ is cofinal, since the $1$ -skeleton of $\Lambda (\kappa )$ is strongly connected. From Theorem 4.4 it follows that $C^{\star }(\kappa )$ is simple.

Now, let $A \in \Lambda (\kappa )^{\mathbf {0}}$ be an arbitrary vertex. Since each edge of the $1$ -skeleton of $\Lambda (\kappa )$ is bidirectional, we can set $\mu $ to be a path that begins at A and traverses a single solid edge to some vertex B, before immediately returning to A. Then $d({\kern1.3pt}\mu ) = (2,0)$ , and, since $\alpha , \beta \geq 3$ , B is the range of some other solid edge, and so $\mu $ is a cycle with an entrance. Then, by strong-connectedness, the conditions of Theorem 4.5 are satisfied, and $\Lambda (\kappa )$ is purely infinite.

In [Reference Evans2] it is shown that, given a row-finite k-rank graph $\Lambda $ with no sources, the $C^{\star }$ -algebra $C^{\star }(\Lambda )$ is separable, nuclear, unital, and satisfies the universal coefficient theorem. Furthermore, we have shown in Proposition 4.6 that, given a complete bipartite graph $\kappa = \kappa (\alpha , \beta )$ with $\alpha , \beta \geq 3$ , the $C^{\star }$ -algebra $C^{\star }(\kappa )$ associated to its $2$ -rank graph is simple and purely infinite. Hence, we can deduce the following result.

Corollary 4.7. Consider the complete bipartite graph $\kappa = \kappa (\alpha ,\beta )$ for $\alpha ,\beta \geq 3$ , with corresponding $2$ -rank graph $\Lambda (\kappa )$ . Then the isomorphism class of the associated $C^{\star }$ -algebra $C^{\star }(\Lambda (\kappa ))$ is completely determined by the K-groups $K_0 (C^{\star }(\kappa )) = K_1 (C^{\star }(\kappa ))$ and the position of the class of the identity in $K_0(C^{\star }(\kappa ))$ .

5 Unpointed tiles

There is an alternative way we could have defined the adjacency matrices above, giving rise to a different $2$ -rank graph structure.

Define an unpointed tile system $(G,U,V,\mathcal {S}')$ in the same way as in Definition 2.4, but replacing $\mathcal {S}=\mathcal {S}(G)$ with the set of unpointed tiles $\mathcal {S}'=\mathcal {S}'(G)$ . We show that analogues of the results in Section 3 also hold for unpointed tile systems.

Definition 5.1. Let $(G, U, V, \mathcal {S}')$ be an unpointed tile system, and let $A', B' \in \mathcal {S}'$ be unpointed tiles, that is, equivalence classes of some pointed tiles $A, B \in \mathcal {S}$ (see Definition 2.3). Recall the matrices $M_1,M_2$ from Definition 3.1. We define functions $M^{\prime }_1,M^{\prime }_2:\mathcal {S}' \times \mathcal {S}' \rightarrow \lbrace 0,1\rbrace $ as follows:

$$ \begin{align*} \begin{array}{r} M^{\prime}_1(A',B') = \left\{ \begin{array}{@{}l} 1 \quad \text{if}\ M_1(A_\bullet ,B_\bullet) = 1,\ \text{for some}\ A_\bullet \sim A,\ B_\bullet \sim B, \\ 0 \quad \text{otherwise,} \end{array} \right.\\ \text{}\\ M^{\prime}_2(A',B') = \left\{ \begin{array}{@{}l} 1 \quad \text{if}\ M_2(A_\bullet ,B_\bullet) = 1,\ \text{for some}\ A_\bullet \sim A, B_\bullet \sim B, \\ 0 \quad \text{otherwise.} \end{array} \right. \end{array} \end{align*} $$

We define adjacency matrices $M^{\prime }_1$ , $M^{\prime }_2$ accordingly.

Proposition 5.2. Consider the complete bipartite graph $\kappa = \kappa (\alpha ,\beta )$ on $\alpha \geq 2$ white and $\beta \geq 2$ black vertices, and let $(\kappa , U, V, \mathcal {S}'(\kappa ))$ be an unpointed tile system. Then the corresponding adjacency matrices $M^{\prime }_1$ and $M^{\prime }_2$ commute, and $(\kappa , U, V, \mathcal {S}'(\kappa ))$ has the UCE property. Hence, $(\kappa ,U,V,\mathcal {S}'(\kappa ))$ has a $2$ -rank graph structure.

Proof. Given two unpointed tiles $A',B' \in \mathcal {S}'(\kappa )$ , consider their respective sets of pointed tiles $\mathcal {A},\mathcal {B} \in \mathcal {S}(\kappa )$ as defined in Definition 5.1. Notice that $M^{\prime }_1(A',B') = 1$ if and only if, for every $A_\bullet \in \mathcal {A}$ , we can find some $B_\bullet \in \mathcal {B}$ such that $M_1 (A_\bullet ,B_\bullet ) = 1$ . The same is true for $M^{\prime }_2$ . Write $A' = ( u_i^1, v_j^1, u_i^2, v_j^2 )$ , and define sets

$$ \begin{align*} X_A := \lbrace T \in \mathcal{S}'(\kappa) \mid M^{\prime}_1(A,T) = 1\rbrace, \quad Y_A := \lbrace T \in \mathcal{S}'(\kappa) \mid M^{\prime}_2(A,T) = 1\rbrace. \end{align*} $$

Then $X_A$ contains precisely those tiles of the form $( u_k^1, v_j^1, u_k^2, v_j^2)$ , where $k \neq i$ , and $Y_A$ only those of the form $( u_i^1, v_l^1, u_i^2, v_l^2)$ , where $l \neq j$ . The proof then proceeds in a similar fashion to that of Proposition 3.3, and the $2$ -rank graph structure follows immediately from [Reference Kumjian and Pask6, Section 6] as in Theorem 3.5.

We write $\Lambda '(\kappa )$ for the $2$ -rank graph induced by the adjacency matrices $M^{\prime }_1$ and $M^{\prime }_2$ . It is not difficult to verify that $\Lambda '(\kappa )$ is row-finite, with a finite vertex set and no sources. Hence, we can apply Theorem 3.8, and derive the following result.

Theorem 5.3 (K-groups for unpointed tile systems)

Let $a,b \geq 0$ , and let ${\kappa (a+2,b+2)}$ be the complete bipartite graph on $a+2$ white and $b+2$ black vertices. Again, without loss of generality, we can assume that $a\leq b$ . Write $C^{\star }(\kappa ) := C^{\star }(\Lambda '(\kappa ))$ . Then, for $\epsilon = 0,1$ , we have the following assertions.

  1. (1) If $a=b=0$ , then $K_\epsilon (C^{\star }(\kappa (a+2,b+2)))=K_\epsilon (C^{\star }(\kappa (2,2))) \cong \mathbb {Z}^2$ .

  2. (2) If $a=0$ and $b \geq 1$ , then

    $$ \begin{align*} K_\epsilon(C^{\star}(\kappa(a+2,b+2))) \cong ( \mathbb{Z}/2)^b\oplus ( \mathbb{Z}/(2b)). \end{align*} $$
  3. (3) If $a,b \geq 1$ , then

    $$ \begin{align*} K_\epsilon(C^{\star}(\kappa(a+2,b+2))) \cong ( \mathbb{Z}/2)^{(a+1)(b+1)-1}\oplus ( \mathbb{Z}/2g), \end{align*} $$
    where $g := \gcd (a, b)$ .

Proof. Again, we start by proving (3) as the first two cases follow. Write $\alpha := a+2$ , $\beta := b+2$ , and let $\alpha , \beta \geq 3$ . For $1 \leq i \leq \alpha $ , $1 \leq j \leq \beta $ , write $A_{ij}'$ for the unpointed tile $( u_i^1, v_j^1, u_i^2, v_j^2 ) \in \mathcal {S}'(\kappa )$ . Then

(5-1) $$ \begin{align} \operatorname{\mathrm{coker}} &= \operatorname{\mathrm{coker}}(\mathbf{1}-(M_1')^T, \mathbf{1}-(M_2')^T )\nonumber\\ &= \bigg\langle A_{ij}' \in \mathcal{S}'(\kappa) \bigg\vert\, A_{ij}' = \sum_{T' \in \mathcal{S}'(\kappa)} M_1'(A_{ij}',T') \cdot T' = \sum_{T' \in \mathcal{S}'(\kappa)} M_2'(A_{ij}',T')\cdot T' \bigg\rangle. \end{align} $$

Fix $p \in \lbrace 1,\ldots , \alpha \rbrace $ , $q \in \lbrace 1,\ldots , \beta \rbrace $ , and notice that:

  • $M^{\prime }_1(A_{pq}', T') = 1$ if and only if $T' = A_{iq}'$ for some $i \neq p$ ;

  • $M^{\prime }_2(A_{pq}', T') = 1$ if and only if $T' = A_{pj}'$ for some $j \neq q$ .

Hence, the relations of (5-1) are equivalent to $A_{pq}' = \sum _{i \neq p} A_{iq}' = \sum _{j \neq q} A_{pj}'$ . Define

$$ \begin{align*} J_{pq} := \bigg(\sum_{i=2}^{\alpha} A_{iq}'\bigg) - A_{pq}' \quad \text{and}\quad I_{pq} := \bigg(\sum_{j=2}^{\beta} A_{pj}'\bigg) - A_{pq}' \end{align*} $$

for $p, q \geq 2$ . Then

$$ \begin{align*} 2J_{pq} &= 2\bigg(\sum_{i=2}^{\alpha} A_{iq}'\bigg) - 2A_{pq}' \\ &= 2(A_{2q}' + \cdots + A_{\alpha q}' - A_{pq}') + A_{1q}' - A_{1q}' \\ &= (A_{1q}' + A_{2q}' + \cdots + A_{\alpha q}' - A_{pq}') + (-A_{1q}' + A_{2q}' + \cdots + A_{\alpha q}') - A_{pq}' \\ &= A_{pq}' + 0 - A_{pq}' = 0, \end{align*} $$

and, similarly, $2I_{pq} = 0$ . Now $J_{pq} = 0$ or $I_{pq} = 0$ only if $A_{pq}' = A_{1q}'$ or $A_{pq}' = A_{p1}'$ , respectively. But, since $\alpha , \beta \geq 3$ , these equivalences are not relations listed at (5-1), and so $\operatorname {\mathrm {ord}}(J_{pq}) = \operatorname {\mathrm {ord}}(I_{pq}) = 2$ . Notice that we can write each $A_{1q}'$ and $A_{p1}'$ in terms of the other $A_{ij}'$ for $p,q \geq 2$ ; hence, we can remove these from the list of generators by a sequence of Tietze transformations.

Also notice that $A_{2q}' = J_{2q} - \sum _{i=3}^\alpha A_{iq}'$ . Proceeding inductively, we can write each $A_{pq}'$ in terms of the $J_{iq}$ and the $A_{iq}'$ for $i> p$ . Similarly, we can express each $A_{pq}'$ in terms of the $I_{pj}$ and the $A_{pj}'$ for $j> q$ . Hence, we can rewrite the generators of $\operatorname {\mathrm {coker}}$ as $A_{11}'$ , $I_{pq}$ , $J_{pq}$ for $p,q \geq 2$ . But $A_{11}' = -(A_{p1}'+J_{p1}) = -(A_{1q}'+I_{1q})$ for all $p,q \geq 2$ , so

$$ \begin{align*} (\alpha - 2)A_{11}' = -\sum_{i=3}^{\alpha} (A_{i1}' + J_{i1}) = -\bigg( J_{21} + \sum_{i=3}^\alpha J_{i1}\bigg ), \end{align*} $$

and $2(\alpha - 2)A_{11}' = 0$ . Similarly, $2(\beta - 2)A_{11}' = 0$ , and, hence, $2gA_{11}' = 0$ , where $g:= \gcd (\alpha -2,\beta -2)$ .

Observe that, since $I_{pq}$ is defined in terms of the $A_{pj}'$ , and each $A_{pj}'$ can be written in terms of the $J_{ij}$ , we can remove the $I_{pq}$ from the list of generators of $\operatorname {\mathrm {coker}}$ . Finally, we can rewrite (5-1) as

$$ \begin{align*} \operatorname{\mathrm{coker}} &= \langle J_{2q}, J_{p2}, J_{pq}, A_{11}' \mid 2J_{2q} = 2J_{p2} = 2J_{pq} = 2gA_{11}' = 0, \\ &\qquad \text{ for } 3 \leq p \leq \alpha, 3 \leq q \leq \beta \rangle, \end{align*} $$

and, after substituting $a = \alpha - 2$ , $b = \beta - 2$ , this gives a presentation for $(\mathbb {Z}/2)^{(a+1)(b+1)-1}\oplus (\mathbb {Z}/2g)$ . There is no torsion-free part, so this proves (3). If $\alpha = 2$ , then $A_{1q}' = A_{2q}'$ for all $1 \leq q \leq \beta $ , so we can write

$$ \begin{align*} \operatorname{\mathrm{coker}} = \bigg\langle A_{1q}' \bigg\vert\, A_{1q}' = \sum_{j\neq q} A_{1j}', \text{ for } 1 \leq q \leq \beta \bigg\rangle. \end{align*} $$

We adjust the proof above accordingly to obtain the result of (2). Finally, in case (1), where $\alpha = \beta = 2$ , we have $A_{11}' = A_{12}' = A_{21}' = A_{22}'$ with no further relations, so that $\operatorname {\mathrm {coker}} = \langle A_{11}' \rangle \cong \mathbb {Z}$ , and the result follows from Theorem 3.8.

Theorem 5.4. Let $\alpha , \beta \geq 3$ , let $\kappa = \kappa (\alpha , \beta )$ be the complete bipartite graph on $\alpha $ white and $\beta $ black vertices, and write $g := \gcd (\alpha - 2, \beta - 2)$ . Then the order of the class of the identity $[\mathbf {1}]$ in $K_0(C^{\star }(\Lambda '(\kappa )))$ is equal to g if g is odd, and $g/2$ if g is even.

Proof. We use the same notation as in the proof of Theorem 5.3. As with Theorem 3.12, we know that the order of $[\mathbf {1}]$ in $K_0(C^{\star }(\kappa ))$ is equal to the order of the sum of all tiles $A_{ij}'$ . We write $\Sigma $ for this sum.

We have $A_{pq}' = \sum _{i \neq p} A_{iq}' = \sum _{j \neq q} A_{pj}'$ , and so $\Sigma = (\alpha - 1)\Sigma = (\beta - 1)\Sigma $ . From this, it follows that $g\Sigma = 0$ . We also have $A_{pq}' = \sum _{i \neq p} \sum _{j \neq q} A_{ij}'$ , so

$$ \begin{align*} \Sigma = A_{pq}' + \sum_{i \neq p} A_{iq}' + \sum_{j \neq q} A_{pj}' + \sum_{i \neq p} \sum_{j \neq q} A_{ij}' = 4A_{pq}' \end{align*} $$

for any fixed $p,q$ . But $2gA_{pq}' = 0$ , and so, if $g=2h$ for some integer h, then $h\Sigma = 4hA_{pq}' = 0$ , and we are done.

The proof of the next proposition is analogous to that of Proposition 4.6.

Proposition 5.5. Consider the complete bipartite graph $\kappa = \kappa (\alpha ,\beta )$ for $\alpha ,\beta \geq 3$ , and the associated $2$ -rank graph $\Lambda '(\kappa )$ . Then the isomorphism class of the universal $C^{\star }$ -algebra $C^{\star }(\Lambda '(\kappa ))$ is completely determined by its K-theory and the position of the class of the identity in $K_0(C^{\star }(\Lambda '(\kappa )))$ .

6 The homology of a tile complex

Theorem 6.1. Let $\kappa = \kappa (\alpha ,\beta )$ be the complete bipartite graph on $\alpha \geq 2$ white and $\beta \geq 2$ black vertices, let $(\kappa , U, V, \mathcal {S}'(\kappa ))$ be an unpointed tile system, and let $TC(\kappa )$ be its associated tile complex. Then the homology groups of $TC(\kappa )$ are given by

$$ \begin{align*} H_n(TC(\kappa)) \cong \begin{cases} 0 & \text{ for }n = 0, \\ \mathbb{Z}^{\alpha + \beta - 2} & \text{ for } n = 1, \\ \mathbb{Z}^{(\alpha - 1)(\beta - 1)} & \text{ for } n = 2, \\ 0 & \text{ for } n \geq 3. \end{cases} \end{align*} $$

Proof. As $TC(\kappa )$ is a path-connected, two-dimensional CW-complex by construction, $H_n(TC(\kappa )) \cong 0$ for $n=0$ and $n \geq 3$ .

The proof is based on that of [Reference Norledge, Thomas and Vdovina7, Proposition 3]. The boundary of each square in $TC(\kappa )$ is given by an element of $\mathcal {S}'(\kappa )$ ; write these elements as $(u_i^1,v_j^1,u_i^2,v_j^2)$ . By construction, $TC(\kappa )$ has four vertices: each is the origin of all directed edges labelled $u_i^1$ , $v_j^1$ , $u_i^2$ and $v_j^2$ . Each tile is homotopy equivalent to a point; pick tile $(u_1^1,v_1^1,u_1^2,v_1^2)$ and contract it, thereby identifying the four vertices. Call the resulting tile complex $TC_1(\kappa )$ . This is a two-dimensional CW-complex whose edges are loops, and whose $2$ -cells comprise for $2 \leq i \leq \alpha $ , $2 \leq j \leq \beta $

  • $(\alpha -1)(\beta -1)$ -many unpointed tiles $A_{ij}'= (u_i^1,v_j^1,u_i^2,v_j^2)$ ;

  • $(\alpha -1)$ -many $2$ -gons $X_i'$ with boundaries described analogously by $(u_i^1,u_i^2)$ ;

  • $(\beta -1)$ -many $2$ -gons $Y_j'$ with boundaries described by $(v_j^1,v_j^2)$ .

Consider the chain complex associated to $TC_1(\kappa )$ :

$$ \begin{align*} \cdots \longrightarrow C_3 \overset{\partial_3}{\longrightarrow} C_2 \overset{\partial_2}{\longrightarrow} C_1 \overset{\partial_1}{\longrightarrow} C_0 \overset{\partial_0}{\longrightarrow} 0. \end{align*} $$

Since $TC_1(\kappa )$ is two-dimensional and has one vertex, this boils down to

$$ \begin{align*} 0 \overset{0}{\longrightarrow} C_2 \overset{\partial_2}{\longrightarrow} C_1 \overset{0}{\longrightarrow} 0, \end{align*} $$

and so $H_1(TC_1(\kappa )) \cong C_1 / \operatorname {\mathrm {im}}(\partial _2)$ and $H_2(TC_1(\kappa )) \cong \ker (\partial _2)$ . We have $\partial _2(A_{ij}') = u_i^1 + v_j^1 + u_i^2 + v_j^2$ , $\partial _2(X_i') = u_i^1 + u_i^2$ , and $\partial _2(Y_j') = v_j^1 + v_j^2$ . Clearly $\ker (\partial _2)$ is generated by $\lbrace A_{ij}' - X_i' - Y_j' \mid 2 \leq i \leq \alpha , 2 \leq j \leq \beta \rbrace $ , which implies that $\ker (\partial _2) \cong \mathbb {Z}^{(\alpha - 1 )( \beta - 1)}$ .

Similarly, an Abelian group presentation for $H_1(TC_1(\kappa ))$ is given by

$$ \begin{align*} H_1(TC_1(\kappa))\cong \langle &u_i^1,v_j^1,u_i^2,v_j^2 \vert\, u_i^1 + v_j^1 + u_i^2 + v_j^2 = u_i^1 + u_i^2 \\ &= v_j^1 + v_j^2 = 0 \text{ for } 2 \leq i \leq \alpha, 2 \leq j \leq \beta \rangle, \end{align*} $$

which, after substituting $u_i^2 = -u_i^1$ and $v_j^2 = -v_j^1$ , gives

$$ \begin{align*} H_1(TC_1(\kappa)) \cong \langle u_i^1,v_j^1 \vert\ 2 \leq i \leq \alpha, 2 \leq j \leq \beta \rangle. \end{align*} $$

This is a presentation for $\mathbb {Z}^{\alpha + \beta - 2}$ , and, since $TC_1(\kappa )$ is homotopy equivalent to $TC(\kappa )$ , we are done.

7 Pointed and unpointed $2t$ -gon systems

In this section we suggest generalisations of the methods above for constructing $C^{\star }$ -algebras associated to $2t$ -gon systems, both for even and arbitrary $t \geq 1$ .

When $t=2$ , we have an innate idea of what it means for two $2t$ -gons to be ‘stackable’ – functions we called horizontal and vertical adjacency in Definition 3.1. We extend this notion to all even $t \geq 2$ in as natural a way as possible. The following definition directly generalises those at the beginning of Section 2.

Definition 7.1. Let G be a connected bipartite graph on $\alpha $ white and $\beta $ black vertices. Let U, V be sets with $|U| = 2t\alpha $ , $|V| = 2t\beta $ , gifted with fixed-point-free involutions $u \mapsto \bar {u}$ , $v \mapsto \bar {v}$ , respectively. Using U and V, construct the $2t$ -polyhedron $P(G)$ from Theorem 2.2, which comprises $2t$ -gons $A_e = [ u_p^1,v_q^1,\ldots , u_p^t,v_q^t ]$ and has G as its link at each vertex. Write $\mathcal {S}'(G) := \lbrace A_e \mid e \in E(G)\rbrace $ for the set of $2t$ -gons comprising $P(G)$ . We call elements of $\mathcal {S}_t'(G)$ unpointed $2t$ -gons, and denote them by $A_e = ( x_1,y_1,\ldots , x_t,y_t)$ .

Analogously to Section 2, we write $[ x_1,y_1,\ldots , x_t,y_t ]$ for a pointed $2t$ -gon, that is, a $2t$ -gon labelled anticlockwise and starting from a distinguished basepoint by the sequence $x_1, y_1, \ldots , x_t, y_t$ , for some $x_i \in U$ , $y_i \in V$ . Write $\mathcal {S}_t=\mathcal {S}_t(G)$ for the set of $2t\alpha \beta $ pointed $2t$ -gons. We call the tuple $(G,U,V,\mathcal {S}_t)$ a $2t$ -gon system. Similarly, we call the tuple $(G,U,V,\mathcal {S}_t')$ an unpointed $2t$ -gon system.

Consider the adjacency matrices $M_1$ and $M_2$ from Definition 3.1. We can view two pointed tiles ( $4$ -gons) $A=[x_1,y_1,x_2,y_2]$ and B as being horizontally adjacent, that is, $M_1(A,B)=1$ , if and only if, after reflecting A through an axis connecting the midpoints of $x_1$ and $x_2$ , and then replacing $x_1$ , $x_2$ by some $x_1' \neq x_1$ , $x_2' \neq x_2$ respectively, we can obtain B. Likewise, if and only if we can obtain B by reflecting A through an axis joining the midpoints of the y edges, and then changing the labels of those edges, do we say that A and B are vertically adjacent.

Definition 7.2. Let t be an even integer, let $(G,U,V,\mathcal {S}_t)$ be a $2t$ -gon system, and let $A = [ x_1,y_1,\ldots ,x_t,y_t] \in \mathcal {S}_t$ be a pointed $2t$ -gon.

Reflect A through an axis joining the midpoints of the sides labelled $x_1$ and $x_{(t/2)+1}$ to obtain a new pointed $2t$ -gon $[ \bar {x}_1, \bar {y}_t, \bar {x}_t, \bar {y}_{t-1}, \ldots , \bar {x}_2, \bar {y}_1]$ . A pointed $2t$ -gon $B \in \mathcal {S}_t$ is V-adjacent to A whenever $B = [ \bar {x}_1', \bar {y}_t, \bar {x}_t', \bar {y}_{t-1}, \ldots , \bar {x}_2', \bar {y}_1 ]$ for some $x_i' \neq x_i$ .

Similarly, reflect A so that $x_1 \mapsto \bar {x}_{(t/2)+1}$ ; we obtain a new pointed $2t$ -gon

(7-1) $$ \begin{align} [ \bar{x}_{(t/2)+1}, \bar{y}_{t/2}, \bar{x}_{t/2}, \ldots , \bar{y}_1, \bar{x}_1, \bar{y}_t, \bar{x}_t, \ldots , \bar{x}_{(t/2)+2}, \bar{y}_{(t/2)+1} ]. \end{align} $$

We say that a pointed $2t$ -gon $B \in \mathcal {S}_t$ is U-adjacent to A if B is of the form (7-1), but with all elements $y_i$ replaced with some $y_i' \neq y_i$ (Figure 9).