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 D = {0, v, 2v, · · · , (|q|−1)v} ⊂ Z2. Let f(x) = x2+px+q be the characteristic polynomial of A. We show that the tile T is disklike if and only if 2|p| ≤ |q+2|. 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.