Combinatorial classification of $(pm 1)$-skew projective spaces

The noncommutative projective scheme $operatorname{mathsf{Proj_{nc}}} S$ of a $(pm 1)$-skew polynomial algebra $S$ in $n$ variables is considered to be a $(pm 1)$-skew projective space of dimension $n-1$. In this paper, using combinatorial methods, we give a classification theorem for $(pm 1)$-skew projective spaces. Specifically, among other equivalences, we prove that $(pm 1)$-skew projective spaces $operatorname{mathsf{Proj_{nc}}} S$ and $operatorname{mathsf{Proj_{nc}}} S$ are isomorphic if and only if certain graphs associated to $S$ and $S$ are switching (or mutation) equivalent. We also discuss invariants of $(pm 1)$-skew projective spaces from a combinatorial point of view.