Non-abelian tensor product and circular orderability of groups

For a group $G$ we consider its tensor square $G\otimes G$ and exterior square $G\wedge G$. We prove that for a circularly orderable group $G$, under some assumptions on $H_1(G)$ and $H_2(G)$, its exterior square and tensor square are left-orderable. This yields an obstruction for a circularly orderable group $G$ to have torsion. We apply these results to study circular orderability of tabulated virtual knot groups.