# Topology of the nodal set of random equivariant spherical harmonics on $S^3$

## TL;DR

This paper investigates the topology of nodal sets of random equivariant spherical harmonics on the 3-sphere, revealing that they typically have a single component and their genus scales cubically with the degree.

## Contribution

It provides a probabilistic analysis of the nodal topology of equivariant spherical harmonics, including expected genus calculations and nodal component counts.

## Key findings

- Almost surely a single nodal component
- Expected genus proportional to m((N^2 - m^2)/2 + N)
- Genus scales as order N^3 when m/N is fixed

## Abstract

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 $N^3$.