Let $n>2$ be an integer. We consider $n$ pairs $(x_1,y_1),\dotsc,(x_n,y_n)$ in $\mathbb{N}^2$, 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 $(x_1,x_1)$. (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 translates of the polygon defined by these points. We let $T(n)\subseteq \mathbb{N}^{2n}$ be the set of all $n$-sets of pairs which define a tileable polygon.

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

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