 Solution

In the Rational map approximation for Skyrmions, stereographic coordinate $z$ describes the direction $\hat{\mathbf{x}}\in S^2$, and a degree-$B$ rational map $R(z)=p(z)/q(z)$ defines a unit vector ${\mathbf{n}}_R$. The Skyrme model field is approximated byIts baryon number is the degree $B$ of $R$. Angular integration reduces the energy to a radial variational problem,up to the conventional overall normalization, where the angular functional $\mathcal{I}$ depends only on $R$. One first minimizes $\mathcal{I}$ among degree-$B$ maps and then minimizes over the profile $f$. This efficiently captures the topology, energy, and polyhedral symmetries of many Skyrmions.

Let $\omega =e^{2\pi i/5}$. Since ${\omega }^5=1$,which is a fivefold spatial rotation accompanied by a target-space rotation. The real coefficients also give $R(\bar{z})=\overline{R(z)}$, while direct substitution givesTogether these transformations extend the cyclic symmetry to the stated $D_{5d}$ symmetry.

For $p=z^7-7z^2$ and $q=7z^5+1$, the Wronskian isBesides $z=0$, put $y=z^5$. ThenThus five zeros lie on the circleat arguments $2\pi k/5$, and five lie on the reciprocal circle at arguments $(2k+1)\pi /5$. The polynomial has degree eleven, so the twelfth zero lies at $z=\infty$. On the Riemann sphere, the zeros therefore form two opposite poles and two staggered pentagonal rings: the twelve vertices of an icosahedron.

The angular baryon-density factor is proportional to $|dR/dz{|}^2$ and vanishes at these critical directions. The Wronskian zeros therefore point toward twelve holes in the baryon-density surface. They are the face centers of the dodecahedral $B=7$ Skyrmion, equivalently the vertices of its dual icosahedron, and make its icosahedral symmetry visible directly in the rational map.