Classification, Koszulity and Artin-Schelter Regularity of certain Graded Twisted Tensor Products

Let $\mathbb{k}$ be an algebraically closed field. We classify all of the quadratic twisted tensor products $A \otimes_{\tau} B$ in the cases where $(A, B) = (\mathbb{k}[x], \mathbb{k}[y])$ and $(A, B) = (\mathbb{k}[x, y], \mathbb{k}[z])$. We determine when a quadratic twisted tensor product of this form is Koszul, and when it is Artin-Schelter regular.