  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 $N^3$.
