On the geometry of Tensor Network States of $2\times N$ Grids

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.