The cobordism class of the moduli space of polygons in ${\mathbb{R}}^3$

For any vector $r=(r_1,\mathrm{.}\mathrm{.}\mathrm{.},r_n)$, let $M_r$ denote the moduli space (under rigid motions) of polygons in ${\mathbb{R}}^3$ with $n$-sides whose lengths are $r_1,\mathrm{.}\mathrm{.}\mathrm{.},r_n$. We give an explicit characterization of the oriented $S^1$-cobordism class of $M_r$ which depends uniquely on the length vector $r$.