We show that real and imaginary parts of equivariant spherical harmonics on $S^3$ have almost surely a single nodal component. Moreover, if the degree of the spherical harmonic is $N$ and the equivariance degree is $m$, then the expected genus is proportional to $m \left(\frac{N^2 - m^2}{2} + N\right) $. Hence if $\frac{m}{N}= c $ for fixed $0 < c < 1$, the genus has order $N^3$.

中文翻译：

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𝕊3 上随机等变球谐函数节点集的拓扑

我们表明，$S^3$ 上等变球谐函数的实部和虚部几乎肯定具有单个节点分量。此外，如果球谐函数的度数为 $N$，等方差度数为 $m$，那么期望的属与 $m \left(\frac{N^2 - m^2}{2} + N \右）$。因此，如果 $\frac{m}{N}= c $ 对于固定的 $0 < c < 1$，则属有阶 $N^3$。