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.