Abstract
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 \({\cal D} \subset {\mathbb{Z}^3}\) be a digit set containing |det M| elements. Then the unique nonempty compact set \(T = T(M,{\cal D})\) defined by the set equation \(MT = T + {\cal D}\) is called an integral self-affine tile if its interior is nonempty. If \({\cal D}\) is of the form \({\cal D} = \{ 0,v, \ldots ,(|\det M| - 1)v\} 0\), 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.
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Acknowledgements
The first author was supported by a grant funded by the Austrian Science Fund and the Russian Science Foundation (Grant No. I 5554). The second author was supported by National Natural Science Foundation of China (Grant No. 12101566). We thank the referees for their suggestions.
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Thuswaldner, J.M., Zhang, SQ. On self-affine tiles that are homeomorphic to a ball. Sci. China Math. 67, 45–76 (2024). https://doi.org/10.1007/s11425-022-2065-2
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DOI: https://doi.org/10.1007/s11425-022-2065-2