  We discuss the geometry of a class of tensor network states, called projected
entangled pair states in the Physics literature. We provide initial results
towards a question of Verstraete and Rizzi regarding the tensor network state
of an $M \times N$ grid; we partially answer the question for a $2 \times N$
grid. We also study the $2 \times N$ grid sitting on a torus and provide
initial results towards understanding the Zariski closure of the set of tensor
network states associated to this graph. Finally, we give explicit tensors that
provide optimal (using currently available methods) bounds on the border rank
of a generic tensor in the tensor network state of the $2 \times N$ grid.
