Connectedness of planar self-affine sets associated with non-collinear digit sets

We study the connectedness of the planar self-affine sets $T(A,\mathcal{D})$ generated by an integer expanding matrix $A$ with $\mathrm{\mid }\det (A)\mathrm{\mid }=3$ and a non-collinear digit set $\mathcal{D}=\{0,v,kAv\}$ where $k\in \mathbb{Z}\setminus \{0\}$ and $v\in {\mathbb{Z}}^2$ such that $\{v,Av\}$ is linearly independent. By checking the characteristic polynomials of $A$ case by case, we obtain a criterion concerning only $k$ to determine the connectedness of $T(A,\mathcal{D})$.