Disklikeness of planar self-affine tiles

We consider the disklikeness of the planar self-affine tile $T$ generated by an integral expanding matrix $A$ and a consecutive collinear digit set ${\mathcal{D}}= \{0, v, 2v, \cdots, (\vert q\vert-1)v \}\subset {\Bbb{Z}}^2$. Let $f(x)=x^{2}+ p x+ q$ be the characteristic polynomial of $A$. We show that the tile $T$ is disklike if and only if $2\vert p\vert\leq \vert q+2\vert$. Moreover, $T$ is a hexagonal tile for all the cases except when $p=0$, in which case $T$ is a square tile. The proof depends on certain special devices to count the numbers of nodal points and neighbors of $T$ and a criterion of Bandt and Wang (2001) on disklikeness.