On self-affine tiles that are homeomorphic to a ball

Let M be a 3 × 3 integer matrix which is expanding in the sense that each of its eigenvalues is greater than 1 in modulus and let D⊂ℤ3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal D} \subset {\mathbb{Z}^3}$$\end{document} be a digit set containing |det M| elements. Then the unique nonempty compact set T=T(M,D)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T = T(M,{\cal D})$$\end{document} defined by the set equation MT=T+D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$MT = T + {\cal D}$$\end{document} is called an integral self-affine tile if its interior is nonempty. If D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal D}$$\end{document} is of the form D={0,v,…,(|detM|−1)v}0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal D} = \{ 0,v, \ldots ,(|\det M| - 1)v\} 0$$\end{document}, 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 ∈ ℤ3} induced by T. This CW complex structure is isomorphic to the CW complex defined by the truncated octahedron.