Let $n>2$ be an integer. We consider $n$ pairs $(x_k, y_k)$ for $k\in\{1,\ldots,n\}$ of members of $\mathbb{N}^2$ and the polygon defined by drawing a straight line from $(x_k, y_k)$ to $(x_{k+1},y_{k+1})$ and from $(x_n, y_n)$ to $(x_1,x_1)$. (The lines may cross.)

We say that the points $(x_1, y_1), \ldots, (x_n,y_n)$ define a tileable polygon, if they are not collinear, and $\mathbb{R}^2$ can be tiled using copies of the polygon defined by these points. We let $T^{2n}\subseteq \mathbb{N}^{2n}$ be the collection of $n$ points defining a tileable polygon.

Question. For $n>2$, is $T^{2n}\subseteq \mathbb{N}^{2n}$ computable?

Apologies. Despite putting in considerable effort to formulate the question, I am not sure it is written in an understandable manner.