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

Abstract

We study the topological properties of a class of planar crystallographic replication tiles. Let $M\in\mathbb{Z}^{2\times2}$ be an expanding matrix with characteristic polynomial $x^2+Ax+B$ ($A,B\in\mathbb{Z}$, $B\geq 2$) and ${\bf v}\in\mathbb{Z}^2$ such that $({\bf v},M{\bf v})$ are linearly independent. Then the equation $$MT+\frac{B-1}{2}{\bf v} =T\cup(T+{\bf v})\cup (T+2{\bf v})\cup \cdots\cup(T+(B-2){\bf v})\cup(-T) $$ 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.