On self-affine tiles that are homeomorphic to a ball

Let M be a 3x3 integer matrix which is expanding in the sense that each of its eigenvalues is greater than 1 in modulus and let D subset of Z(3) be a digit set containing vertical bar det M vertical bar elements. Then the unique nonempty compact set T = T(M, D) defined by the set equation MT = T + D is called an integral self-affine tile if its interior is nonempty. If D is of the form D = {0, nu,..., (vertical bar det M vertical bar- 1)nu}, we say that T has a collinear digit set. The present paper is devoted to the topology of integral self-affine tiles with collinear digit sets. In particular, we prove that a large class of these tiles is homeomorphic to a closed 3-dimensional ball. Moreover, we show that in this case, T carries a natural CW complex structure that is defined in terms of the intersections of T with its neighbors in the lattice tiling {T + z : z is an element of Z(3)} induced by T. This CW complex structure is isomorphic to the CW complex defined by the truncated octahedron.