1 Introduction
In [Reference Vdovina15], it was shown how to construct a twodimensional CWcomplex 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 higherrank 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 CWcomplexes, 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 higherrank graph, and induce a universal $C^{\star }$ algebra: the higherrank graph algebra. We use a result of [Reference Evans2] to calculate the Kgroups of these algebras (Theorems 3.10 and 5.3).
In Section 6, we show that the tile complexes have torsionfree 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 twodimensional complexes consisting entirely of $2t$ gons. In all, we associate $2$ rank graph $C^{\star }$ algebras to five systems, and compute their Ktheory in the following theorems:

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

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

(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 higherrank graphs in order to demonstrate the large intersection between the fields of kgraphs 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 tgons, with directed edges labelled from some set $\mathcal {U}$ . By gluing together likelabelled edges (respecting their direction), we obtain a twodimensional complex P. We call such a complex a tpolyhedron.
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 fixedpointfree 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 forwarddirected 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).
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
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 (21). 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 bottomleft 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 (21) 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.
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 VHdatum, 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:
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.
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) $M_1$ and $M_2$ are symmetric and commute with each other;

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

(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 $A,D \in \mathcal {S}(\kappa )$ .
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 socalled krank graph (specifically a $2$ rank graph), as introduced in [Reference Kumjian and Pask6] to build on work by [Reference Robertson and Steger10].
3.1 Higherrank 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 krank graph (or simply a kgraph) 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 higherrank 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 2rank graph structure.
Proof. Following the method of Theorem 2.2, label the elements of the sets U, V
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:
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, $F^1$ ).
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:

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

(b) $\lambda (p,q) \in E^1$ for some $(p,q) \in W(m1,n)$ ;

(c) $\mu (p,q) \in F^1$ for some $(p,q) \in W(m,n1)$ ;

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

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

(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 $\varphi = \lambda (0,0) \cdots \lambda (m1,0)$ and $\psi = \mu (m,0) \cdots \mu (m,n1)$ . If, instead (or as well), $r_F(\psi ) = s_E(\varphi )$ , then there is a unique element $\psi \varphi $ such that $\varphi = \lambda (0,n) \cdots \lambda (m1,n)$ and $\psi = \mu (0,0) \cdots \mu (0,n1)$ (Figure 6).
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

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

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

(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 krank 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 rowfinite 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 krank graph.
Definition 3.7. Let $\Lambda = (\Lambda ,d)$ be a rowfinite krank 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.

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

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

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

(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 rowfiniteness condition, property (d) is not well defined.
Theorem 3.8 (Evans [Reference Evans2, Proposition 4.4])
Let $\Lambda $ be a rowfinite $2$ graph with no sources, finite vertex set $\Lambda ^{\mathbf {0}}$ , and vertex matrices $M_E$ , $M_F$ . Then
where
$ \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 torsionfree 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
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 rowfinite, has no sources, and satisfies $\Lambda (\kappa )^{\mathbf {0}} = 4\alpha \beta $ , whence the result follows from Theorem 3.8.
Theorem 3.10 (Kgroups 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) 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) 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) If $a,b \geq 2$ and $a,b$ are coprime, then
$$ \begin{align*} K_\epsilon (C^{\star} (\kappa (a+2,b+2))) \cong (\mathbb{Z}/a)^{ba} \oplus ( \mathbb{Z}/ab )^{a+1} \oplus \mathbb{Z}^{2(a+1)(b+1)}. \end{align*} $$ 
(4) If $a,b \geq 2$ and $a,b$ are not coprime, then
$$ \begin{align*} K_\epsilon(C^{\star} (\kappa(a+2,b+2))) \cong (\mathbb{Z}/a)^{ba} \oplus ( \mathbb{Z}/l )^{a+1} \oplus ( \mathbb{Z}/g)^{a+2} \oplus \mathbb{Z}^{2(a+1)(b+1)}, \end{align*} $$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
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 (31) 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
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
We have the following equalities:
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 (32), we find that
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
and, when $\alpha = 3$ and $\beta \geq 3$ ,
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 kgraph $C^{\star }$ 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 Kgroups of this $C^{\star }$ algebra are both $\mathbb {Z}^8$ , in agreement with Theorem 3.10.
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 krank 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 Ktheory. 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
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 krank graph, which we can visualise as a nonnegative integer lattice in $\mathbb {R}^k$ (see Figure 6).
Definition 4.1. Let $\Lambda $ be a krank 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 midcycle 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.
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
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
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 twosided 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 krank 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 krank 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 strongconnectedness, the conditions of Theorem 4.5 are satisfied, and $\Lambda (\kappa )$ is purely infinite.
In [Reference Evans2] it is shown that, given a rowfinite krank 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 Kgroups $K_0 (C^{\star }(\kappa )) = K_1 (C^{\star }(\kappa ))$ 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:
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
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 rowfinite, with a finite vertex set and no sources. Hence, we can apply Theorem 3.8, and derive the following result.
Theorem 5.3 (Kgroups 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) 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) 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) 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
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 (51) are equivalent to $A_{pq}' = \sum _{i \neq p} A_{iq}' = \sum _{j \neq q} A_{pj}'$ . Define
for $p, q \geq 2$ . Then
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 (51), 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
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 (51) as
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 torsionfree part, so this proves (3). If $\alpha = 2$ , then $A_{1q}' = A_{2q}'$ for all $1 \leq q \leq \beta $ , so we can write
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
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 Ktheory 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
Proof. As $TC(\kappa )$ is a pathconnected, twodimensional CWcomplex 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 twodimensional CWcomplex 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 $TC_1(\kappa )$ :
Since $TC_1(\kappa )$ is twodimensional and has one vertex, this boils down to
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
which, after substituting $u_i^2 = u_i^1$ and $v_j^2 = v_j^1$ , gives
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 fixedpointfree 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 $[ \bar {x}_1, \bar {y}_t, \bar {x}_t, \bar {y}_{t1}, \ldots , \bar {x}_2, \bar {y}_1]$ . A pointed $2t$ gon $B \in \mathcal {S}_t$ is Vadjacent to A whenever $B = [ \bar {x}_1', \bar {y}_t, \bar {x}_t', \bar {y}_{t1}, \ldots , \bar {x}_2', \bar {y}_1 ]$ 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
We say that a pointed $2t$ gon $B \in \mathcal {S}_t$ is Uadjacent to A if B is of the form (71), but with all elements $y_i$ replaced with some $y_i' \neq y_i$ (Figure 9).