THE K-THEORY OF THE 
${\mathit{C}}^{\star }$
 -ALGEBRAS OF 2-RANK GRAPHS ASSOCIATED TO COMPLETE BIPARTITE GRAPHS

Abstract 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.


Introduction
In [15], it was shown how to construct a two-dimensional CW-complex whose link at each vertex is a complete bipartite graph. In [6], generalising the work of [10], certain combinatorial objects called higher-rank graphs were defined and then associated with a generalisation of a graph algebra [9,Ch. 1]. We combine these two methods to build an infinite family of C -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 [5]. Since these adjacency matrices commute, they characterise the structure of a higher-rank graph, and induce a universal C -algebra: the higher-rank graph algebra. We use a result of [2] to calculate the K-groups of these algebras (Theorems 3.10 and 5.3).
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 -algebras to five systems, and compute their K-theory 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 [10], 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, α, β are positive integers, and κ(α, β) denotes the complete connected bipartite graph on α white and β black vertices.
2. The tile system associated to a bipartite graph DEFINITION 2.1. Let t ∈ Z satisfy t ≥ 2, and let A 1 , . . . , A n be a sequence of solid t-gons, with directed edges labelled from some set 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 [15]). Let G be a connected bipartite (undirected) graph on α white and β black vertices, with edge set E(G). Then, for each t ≥ 1, we can construct a 2t-polyhedron P(G) which has G as the link at each vertex.
We refer to [15], in which it was shown how to build such a 2t-polyhedron. The general method is as follows.
Write U = {u 1 , . . . , u α } for the set of white vertices of G, and V = {v 1 , . . . , v β } for the set of black vertices.
Let U be a set with 2tα elements, indexed u 1 i , u 2 i , . . . , u t i ,ū 1 i ,ū 2 i , . . .ū t i for each u i ∈ U , and let V be the corresponding set with 2tβ elements. Define fixed-point-free involutions u r i →ū r i and v r i →v r i 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.
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.
Label the boundary of A e anticlockwise, starting from the base, by the sequence giving each side of the boundary a forward-directed arrow. We denote this pointed oriented 2t-gon by . 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 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 S (G) the quotient of S(G) with respect to this relation, and we write A e = (u 1 p , v 1 q , u 2 p , v 2 q ) for the equivalence class of A e in S (G). Then S (G) is the set of geometric squares (that is, disregarding basepoint and orientation) in TC(G). We call elements of 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 S(G) ⊆ U × V × U × 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 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 α white and β black vertices. Let U, V be sets with |U| = 4α, |V| = 4β, as constructed above, and let S = S(G) ⊆ U × V × U × V be the corresponding set of pointed tiles. We call the datum (G, U, V, S) a tile system.
, and so on. These four pointed squares represent different pointed tiles, but the same unpointed tile.
x 1ȳ This construction is closely related to, and indeed modelled on, that of a VH-datum, introduced in [16] and developed further in [1].
3. The C -algebra corresponding to a tile system DEFINITION 3.1. Let (G, U, V, 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 S. We define two 4αβ × 4αβ 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, S) be a tile system, and let A, B, C be pointed tiles in S(G) such that M 1 (A, B) = 1 and M 2 (A, C) = 1. We say that the tile system (G, U, V, S)  p , v 2 q , u 1 p , v 1 q ] ∈ S(κ) for some p i, q j. By the completeness of κ and the fact that α, β ≥ 2, there are tiles Similarly, given A, B, C ∈ S(κ) as above, D is the unique tile adjacent to both B and C, and so (κ, U, V, S(κ)) 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 [6] to build on work by [10].
We direct the reader to [14], 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 × 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 [6] 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 κ = κ(α, β) be the complete bipartite graph on α ≥ 2 white and β ≥ 2 black vertices, and let (κ, U, V, S(κ)) be a tile system with adjacency matrices M 1 , M 2 . Then (κ, U, V, S(κ)) has a 2-rank graph structure.
PROOF. Following the method of Theorem 2.2, label the elements of the sets U, V where u 1 , . . . , u α and v 1 , . . . , v β are the white and black vertices of κ, respectively. Construct the tile complex TC(κ), and consider the set S(κ) ⊆ U × V × U × V of pointed tiles of TC(κ). Since κ 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 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: . 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 θ defines a unique dashed-solid path from A to D. The 1-skeleton of the 2-rank graph Λ(κ(α, β)) is strongly connected only when α, β ≥ 3.
can be represented as an m × n grid. The isomorphism θ defines commuting squares. Here is an element of Λ (2,5) .
whenever these conditions make sense. In this way, composition is defined in Λ, and, by construction, we have associativity and the factorisation property of Definition 3.4. Thus Λ, together with obvious degree functor d : (A, λ, μ) → (m, n) for (A, λ, μ) ∈ Λ (m,n) , has the structure of a 2-rank graph, and we write (Λ, d) = Λ(κ). DEFINITION 3.6. Let (Λ, d) be a k-rank graph, let n ∈ N k , and let v ∈ Λ 0 . Write Λ n (v) for the set of morphisms in Λ n which map onto v, that is, As an extension of the concept of a graph algebra (see [9]), we can associate a C -algebra to a k-rank graph. DEFINITION 3.7. Let Λ = (Λ, d) be a row-finite k-rank graph with no sources. We define C (Λ) to be the universal C -algebra generated by a family {s λ | λ ∈ Λ} of partial isometries satisfying the following properties.
Note that, without the row-finiteness condition, property (d) is not well defined. [9] The K-theory of the C * -algebras of 2-rank graphs 127 |Λ 0 | = n, 1 is the n × n identity matrix, ( * , * ) denotes the corresponding block n × 2n matrix, rk(G) is the torsion-free rank of a finitely generated Abelian group G, and tor(G) is the torsion part of G.
(2) If a = 0, 1 and b ≥ 1, then (3) If a, b ≥ 2 and a, b are coprime, then (4) If a, b ≥ 2 and a, b are not coprime, then where (Z/a) 0 is defined to be the trivial group in the case that a = b.
Assume that a, b ≥ 2. Write α = a + 2 and β = b + 2, and for for the tiles with the same edge labels as the horizontal reflection, vertical reflection, and rotation by π of A ij , respectively. Then . Now fix some p ∈ {1, . . . , α} and q ∈ {1, . . . , β}, and notice that: for some j q.
Hence, the relations of (3-1) are equations of the form A pq = i p B iq = j q C pj , and so on for each B pq , C pq , D pq . In particular, we can write B pq = i p A iq and C pq = j q A pj so that Define J q := α i=1 A iq , and I p := β j=1 A pj . Then (α − 2)J q = (β − 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 gΣ = 0, where Σ := 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 = i p j q A ij . Hence, we can remove all the B pq , C pq , and D pq from the list of generators of coker, yielding We have the following equalities: (2, 2)), depicted in Figure 5, consists of four copies of C 2 × C 2 , where C 2 is the cyclic 1-graph with two vertices.
From the diagram, we can see that the (1-skeleton of the) 2-rank graph Λ(κ(2, 2)) comprises four connected components, each being the Cartesian product C 2 × C 2 , depicted in Figure 7. It is well known that the k-graph C -algebra of C 2 is isomorphic to M 2 (C(T)). Furthermore, there is a natural isomorphism C (C m × C n ) M mn (C(T 2 )), and so C (κ(2, 2)) (M 4 (C(T 2 ))) 4 . The K-groups of this C -algebra are both Z 8 , in agreement with Theorem 3.10.
PROOF. From [3], it follows that the order of [1] in K 0 (C (κ)) is equal to the order of the sum of pointed tiles in S(κ); by considerations in the proof of Theorem 3.10, we know this to be g.

Aperiodicity and Kirchberg-Phillips classification
Kumjian and Pask in [6] developed conditions under which the C -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 (κ), and thus, by the Kirchberg and Phillips results [4,8], that the C (κ) are completely classified by their K-theory. We detail the following definitions from [6].
Let k ≥ 1, and let Ω k be the countable small category with object set Ob(Ω k ) := N k 0 and morphism set Hom(Ω k ) given by We identify Ob(Ω k ) with the set of identity morphisms {(m, m) | m ∈ N k 0 }, and, hence, identify Ω k with Hom(Ω k ). Define range and source maps r(m, n) := m and s(m, n) := n, respectively. Then Ω k , together with the degree map d(m, n) := n − m, is a k-rank graph, which we can visualise as a nonnegative integer lattice in R k (see Figure 6).
Given a vertex v ∈ Λ 0 , we write Λ ∞ (v) for the set of infinite paths that begin at v, that is, Let p ∈ Z k , and let ϕ ∈ Λ ∞ . We say that p is a period for ϕ if, for every (m, n) ∈ Ω k with m + p ≥ 0, we have ϕ(m + p, n + p) = ϕ(m, n). We call ϕ periodic if we can find a nonzero period for ϕ.
Given q ∈ N k 0 and a path ϕ ∈ Λ ∞ , we write ϕ q (m, n) := (m + q, n + q). We say that ϕ is eventually periodic if we can find some nonzero q ∈ N k 0 such that ϕ q is periodic. We say that an infinite path ϕ is aperiodic if it is neither periodic nor eventually periodic.
We say that Λ satisfies the aperiodicity condition (also referred to in the literature as Condition (A)) if, for every vertex v ∈ Λ 0 , we can find an aperiodic path ϕ ∈ Λ ∞ (v). We say that Λ is cofinal if, for every vertex v ∈ Λ 0 and every infinite path ϕ ∈ Λ ∞ , we can find λ ∈ Λ and n ∈ N k 0 such that r(λ) = v and s(λ) = ϕ(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. In order to get a feeling as to why this is true, consider Figure 8, which shows a representation of Λ(κ (3, 3)). Each vertex is labelled by a pointed tile from S(κ(3, 3)), and since each tile is vertically adjacent to two others (and horizontally adjacent to [13] The K-theory of the C * -algebras of 2-rank graphs 131 two others), there are two solid arrows and two dashed arrows emanating from each vertex of Λ(κ (3, 3)). This suggests that, analogously to the 1-graph condition, we can always find an entrance to any cycle in Λ, namely by stopping mid-cycle at a vertex, and diverting the path down the second of the two available edges. Hence, as long as α, β ≥ 3, there is enough choice at each vertex to be able to exit a cycle.
PROOF. Firstly, write Λ = Λ(κ), and let A ∈ Λ 0 be an arbitrary vertex. We construct an aperiodic infinite path beginning from A in the following way. Let x : Ω 1 → m≥0 Λ (m,0) be a 1-graph morphism such that x(0) = A. The vertex A represents a pointed tile in S(κ), which is horizontally adjacent to β − 1 other pointed tiles. Hence, A is connected by bidirectional blue arrows to β − 1 other vertices in Λ. Choose two of these vertices, B 1 and B 2 , say, and let x be such that for all m ∈ N 0 . Since x forms an aperiodic sequence, there is no p ∈ Z such that x(m, m) = x(m + p, m + p) for all m, nor any q ∈ N such that x q is periodic; hence, x is an aperiodic path. Similarly, define y : Ω 1 → n≥0 Λ (0,n) by 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 ϕ((m, n), (m, n)) = D precisely when m = r 2 + r + 1 and n = s 2 + s + 1, for some r, s ≥ 1. As above, there is no p ∈ Z 2 such that ϕ((m, n), (m, n)) = ϕ((m, n) + p, (m, n) + p), nor any q ∈ N 2 0 such that ϕ 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 [11,Ch. 5]. DEFINITION 4.3. Let A be a unital C -algebra, and let B ⊂ A be a C -subalgebra. We say that B is hereditary if, for all a, b ∈ A, if b ∈ B and a ≤ b, then a ∈ B.
We say that A is simple if it has no nontrivial closed two-sided ideals.
If A is simple, we say that it is purely infinite if every nonzero hereditary C -subalgebra of A contains a projection which is Murray-von Neumann equivalent to a proper subprojection of itself. Equivalently, A is purely infinite if every nonzero hereditary C -subalgebra contains a projection equivalent to 1.  . Let Λ be a k-rank graph that is cofinal and satisfies the aperiodicity condition. Suppose that, for every v ∈ Λ 0 , we can find λ ∈ Λ with r(λ) = v, and some cycle μ ∈ Λ with an entrance, such that d( μ) 0, and s(λ) = r( μ) = s( μ). Then C (Λ) is purely infinite. PROPOSITION 4.6. Consider κ = κ(α, β) for α, β ≥ 3, and let Λ(κ) be the corresponding 2-rank graph. Then the C -algebra C (κ) from Definition 3.7 is simple and purely infinite.
PROOF. Firstly observe that Λ(κ) is cofinal, since the 1-skeleton of Λ(κ) is strongly connected. From Theorem 4.4 it follows that C (κ) is simple. Now, let A ∈ Λ(κ) 0 be an arbitrary vertex. Since each edge of the 1-skeleton of Λ(κ) is bidirectional, we can set μ 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( μ) = (2, 0), and, since α, β ≥ 3, B is the range of some other solid edge, and so μ is a cycle with an entrance. Then, by strong-connectedness, the conditions of Theorem 4.5 are satisfied, and Λ(κ) is purely infinite.
In [2] it is shown that, given a row-finite k-rank graph Λ with no sources, the C -algebra C (Λ) is separable, nuclear, unital, and satisfies the universal coefficient theorem. Furthermore, we have shown in Proposition 4.6 that, given a complete bipartite graph κ = κ(α, β) with α, β ≥ 3, the C -algebra C (κ) 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 κ = κ(α, β) for α, β ≥ 3, with corresponding 2-rank graph Λ(κ). Then the isomorphism class of the associated C -algebra C (Λ(κ)) is completely determined by the K-groups K 0 (C (κ)) = K 1 (C (κ)) and the position of the class of the identity in K 0 (C (κ)).

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, S ) in the same way as in Definition 2.4, but replacing S = S(G) with the set of unpointed tiles S = S (G). We show that analogues of the results in Section 3 also hold for unpointed tile systems.

. Write
, and define sets Then X A contains precisely those tiles of the form (u 1 , where l j. The proof then proceeds in a similar fashion to that of Proposition 3.3, and the 2-rank graph structure follows immediately from [6, Section 6] as in Theorem 3.5. We write Λ (κ) for the 2-rank graph induced by the adjacency matrices M 1 and M 2 . It is not difficult to verify that Λ (κ) 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 ≥ 0, and let κ(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 ≤ b. Write C (κ) := C (Λ (κ)). Then, for = 0, 1, we have the following assertions.
PROOF. Again, we start by proving (3) as the first two cases follow. Write α := a + 2, β := b + 2, and let α, β ≥ 3. For 1 ≤ i ≤ α, 1 ≤ j ≤ β, write A ij for the unpointed tile Fix p ∈ {1, . . . , α}, q ∈ {1, . . . , β}, and notice that: Hence, the relations of (5-1) are equivalent to A pq = i p A iq = j q A pj . Define The K-theory of the C * -algebras of 2-rank graphs 135 for p, q ≥ 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 α, β ≥ 3, these equivalences are not relations listed at (5-1), and so ord(J pq ) = 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 ≥ 2; hence, we can remove these from the list of generators by a sequence of Tietze transformations. Also notice that A 2q = J 2q − α i=3 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 coker as A 11 , I pq , and 2(α − 2)A 11 = 0. Similarly, 2(β − 2)A 11 = 0, and, hence, 2gA 11 = 0, where g := gcd(α − 2, β − 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 coker. Finally, we can rewrite (5-1) as and, after substituting a = α − 2, b = β − 2, this gives a presentation for (Z/2) (a+1)(b+1)−1 ⊕ (Z/2g). There is no torsion-free part, so this proves (3). If α = 2, then A 1q = A 2q for all 1 ≤ q ≤ β, so we can write We adjust the proof above accordingly to obtain the result of (2). Finally, in case (1), where α = β = 2, we have A 11 = A 12 = A 21 = A 22 with no further relations, so that coker = A 11 Z, and the result follows from Theorem 3.8.
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 [1] in K 0 (C (κ)) is equal to the order of the sum of all tiles A ij . We write Σ for this sum. We have A pq = i p A iq = j q A pj , and so Σ = (α − 1)Σ = (β − 1)Σ. From this, it follows that gΣ = 0. We also have A pq = i p j q A ij , so for any fixed p, q. But 2gA pq = 0, and so, if g = 2h for some integer h, then hΣ = 4hA pq = 0, and we are done.
The proof of the next proposition is analogous to that of Proposition 4.6.
PROOF. As TC(κ) is a path-connected, two-dimensional CW-complex by construction, H n (TC(κ)) 0 for n = 0 and n ≥ 3. The proof is based on that of [7,Proposition 3]. The boundary of each square in TC(κ) is given by an element of S (κ); write these elements as ( . By construction, TC(κ) has four vertices: each is the origin of all directed edges labelled Each tile is homotopy equivalent to a point; pick tile (u 1 1 , v 1 1 , u 2 1 , v 2 1 ) and contract it, thereby identifying the four vertices. Call the resulting tile complex TC 1 (κ). This is a two-dimensional CW-complex whose edges are loops, and whose 2-cells comprise for 2 ≤ i ≤ α, 2 ≤ j ≤ β ; • (α − 1)-many 2-gons X i with boundaries described analogously by (u 1 i , u 2 i ); • (β − 1)-many 2-gons Y j with boundaries described by (v 1 j , v 2 j ).

Pointed and unpointed 2t-gon systems
In this section we suggest generalisations of the methods above for constructing C -algebras associated to 2t-gon systems, both for even and arbitrary t ≥ 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 ≥ 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 α white and β black vertices. Let U, V be sets with |U| = 2tα, |V| = 2tβ, gifted with fixed-point-free involutions u →ū, v →v, respectively. Using U and V, construct the 2t-polyhedron P(G) from Theorem 2.2, which comprises 2t-gons and has G as its link at each vertex. Write S (G) := {A e | e ∈ E(G)} for the set of 2t-gons comprising P(G). We call elements of S t (G) unpointed 2t-gons, and denote them by A e = (x 1 , y 1 , . . . , x t , y t ).
Analogously to Section 2, we write [x 1 , y 1 , . . . , 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 , . . . , x t , y t , for some x i ∈ U, y i ∈ V. Write S t = S t (G) for the set of 2tαβ pointed 2t-gons. We call the tuple (G, U, V, S t ) a 2t-gon system. Similarly, we call the tuple (G, U, V, 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 x 1 , x 2 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, S t ) be a 2t-gon system, and let A = [x 1 , y 1 , . . . , x t , y t ] ∈ S t be a pointed 2t-gon.
Here, the addition in the subscript indices is defined modulo t. Note that each S ∈ S t (κ) takes one of the above forms. Then But, comparing this to (3-1), we see this is precisely a presentation for the direct sum of t/2 copies of coker(I − M T 1 , I − M T 2 ) as in Theorem 3.10, and the result follows. THEOREM 7.5. Let α, β ≥ 3, let t ≥ 2 be even, and let κ = κ(α, β) be the complete bipartite graph on α white and β black vertices. Then the order of the class of the identity [1] in K 0 (C (Λ t (κ))) is equal to g := gcd(α − 2, β − 2).
Furthermore, the isomorphism class of C (Λ t (κ)) is completely determined by the K-groups from Theorem 7.4 and the order of [1] in K 0 .
PROOF. The result follows from Theorems 3.12 and 7.4, and considerations similar to those in the proof of Proposition 4.6.
If we extend the concept of U-and V-adjacency from Definition 7.2 in the obvious way, we can obtain a generalisation of Section 5 for unpointed 2t-gon systems of complete bipartite graphs. Write Λ t (κ) for the induced 2-rank graph. We realise that the proof of Theorem 5.3 does not depend on the number of sides 2t of the 2t-gons; nor do the K-groups associated with Λ t (κ). COROLLARY 7.6 (to Theorem 5.3: K-groups for unpointed 2t-gon systems). Let α, β ≥ 2, and let κ = κ(α, β) be the complete bipartite graph on α white and β black vertices. Then K (C (Λ t (κ))) K (C (Λ (κ))) for = 0, 1, and all t ≥ 1. PROPOSITION 7.7. Let α, β ≥ 3, let κ = κ(α, β) be the complete bipartite graph, and write g := gcd(α − 2, β − 2). Then, for all t ≥ 1, the order of the class of the identity [1] in K 0 (C (Λ t (κ))) is equal to g if g is odd, and g/2 if g is even.
7.1. Questions on canonicality. Corollary 7.6 gives us a collection of K-groups corresponding to systems of 2t-gons with an arbitrary even number of sides 2t, whereas in the pointed case, Theorem 7.4 insists on 2t being divisible by 4. This is due to how we define adjacency in each instance: in the 2t-polyhedron P(κ), each face is adjacent to every other, and since the number of faces is not dependent on t, nor are the U-and V-adjacency matrices in an unpointed 2t-gon system. Adjacency in the pointed case is more difficult to define canonically. When t = 2, and we are dealing with tiles, there is an obvious pair of adjacency functions. We extend these in Definition 7.2, thinking of two 2t-gons as adjacent if we can reflect one horizontally or vertically in order to obtain the form of the other. This works since horizontal and vertical reflections commute, and so the 2t-gon system has the UCE property. If t is not even, then there are no two distinct reflections of 2t-gons which commute, and preserve the structure of pointed 2t-gons. We must pick the same two reflections for both adjacency functions, or else some combination of rotations and identity transformations. None of these options is a direct extension of our horizontal and vertical adjacency functions from Section 3, and so there is no natural choice.
We suggest that the following definitions of U-and V-adjacency for pointed 2t-gons are the most intuitive for t ≥ 3, based on the idea that adjacent 2t-gons should have opposite orientations. They do not, however, generalise the tile systems from Sections 2-5, themselves being the most natural constructions when t = 2. Because of this, the previous constructions are the main focus of this paper. DEFINITION 7.9. Let t ≥ 1 be a fixed arbitrary integer, let (G, U, V, S t ) be a 2t-gon system, and let A = [x 1 , y 1 , . . . , x t , y t ] ∈ S t be a pointed 2t-gon.
A pointed 2t-gon B ∈ S t is V * -adjacent to A if and only if B = [x 1 ,ȳ t , . . . ,x 2 ,ȳ 1 ] for some x i x i .
Similarly, we say that a pointed 2t-gon C ∈ S t is U * -adjacent to A if and only if C = [x 1 ,ȳ t , . . . ,x 2 ,ȳ 1 ] for some y i y i . We define the U * -and V * -adjacency matrices M * U and M * V , respectively, as above.
The proof of the following proposition is almost identical to that of Proposition 7.3, together with Proposition 3.5. From this, along with Theorem 3.8, we can deduce Theorem 7.11.