Topology of planar self-affine tiles with collinear digit set

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We consider the self-affine tiles with collinear digit set defined as follows. Let A,B satisfy |A|≤ B≥ 2 and M²×² be an integral matrix with characteristic polynomial x²+Ax+B. Moreover, let D={0,v,2v,,(B-1)v} for some v² such that v,M v are linearly independent. We are interested in the topological properties of the self-affine tile T defined by MT=(T+d). Lau and Leung proved that T is homeomorphic to a closed disk if and only if 2|A|≤ B+2. In particular, T has no cut point. We prove here that T has a cut point if and only if 2|A|≥ B+5. For 2|A|-B∈ {3,4}, the interior of T is disconnected and the closure of each connected component of the interior of T is homeomorphic to a closed disk.