A Family of Universally Optimal Configurations on Rectangular Flat Tori

We utilize recently introduced linear programming bounds for the energy of periodic configurations in ${\mathbb{R}}^d$ to construct configurations which are universally optimal among those of the form ${\omega }_4+L_{\beta }$, where ${\omega }_4$ is a 4-point multiset and $L_{\beta }\subseteq {\mathbb{R}}^2$ is a rectangular lattice. For two values of the parameter $\beta$, the configurations are obtained from the $A_2$ lattice. Other values of $\beta$ yield the third, and largest cardinality, class of examples of $\mathrm{\Lambda }$-universally optimal configurations which do not arise from lattices $\mathrm{\Lambda }$ known or conjectured to be universally optimal in ${\mathbb{R}}^d$.