Topology of a class of $p2$-crystallographic replication tiles

Abstract

We study the topological properties of a class of planar crystallographic replication tiles. Let $M\in {\mathbb{Z}}^{2\times 2}$ be an expanding matrix with characteristic polynomial $x^2+Ax+B$ ($A,B\in \mathbb{Z}$, $B\ge 2$) and $\mathbf{v}\in {\mathbb{Z}}^2$ such that $(\mathbf{v},M\mathbf{v})$ are linearly independent. Then the equation $$MT+\frac{B-1}{2}\mathbf{v}=T\cup (T+\mathbf{v})\cup (T+2\mathbf{v})\cup \cdots \cup (T+(B-2)\mathbf{v})\cup (-T)$$ defines a unique nonempty compact set $T$ satisfying $\overline{T^o}=T$. Moreover, $T$ tiles the plane by the crystallographic group $p2$ generated by the $\pi$-rotation and the translations by integer vectors. It was proved by Leung and Lau in the context of self-affine lattice tiles with collinear digit set that $T\cup (-T)$ is homeomorphic to a closed disk if and only if $2|A|<B+3$. However, this characterization does not hold anymore for $T$ itself. In this paper, we completely characterize the tiles $T$ of this class that are homeomorphic to a closed disk.

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