 Cubic four-Skyrmion

The familiar low-energy $B=4$ Skyrmion has a cubic baryon-density shell with six face-hole directions and full density symmetry $O_h$, the symmetry group of a cube. The proper subgroup is $O\cong S_4$, the rotational symmetry group of a cube; its field invariance includes compensating isorotations. A useful rational map approximation for Skyrmions has $R(z)=(z^4+2i\sqrt{3}{z}^2+1)/(z^4-2i\sqrt{3}{z}^2+1)$. The relation $R(iz)=1/R(z)$ pairs a spatial quarter-turn with a target half-turn. Its branch directions are $z=0,\infty ,\pm 1,\pm i$, corresponding to the six coordinate-axis directions. The map gives an approximate field, not an exact analytic energy-minimizing solution.