# Topological Properties of Phase Singularities in Wave Fields

## TL;DR

This paper investigates the topological characteristics of phase singularities in wave fields across two and three dimensions, revealing their internal structure, topological charges, and invariants, including knotted configurations.

## Contribution

It introduces a detailed analysis of phase singularities using $$-mapping topological current theory, highlighting their topological charges and invariants in various dimensions.

## Key findings

- Topological inner structures of phase singularities are characterized.
- Topological charges are expressed via Hopf indices and Brouwer degrees.
- Topological invariants of knotted phase singularities are discussed.

## Abstract

Phase singularities as topological objects of wave fields appear in a variety of physical, chemical, and biological scenarios. In this paper, by making use of the $\phi$-mapping topological current theory, we study the topological properties of the phase singularities in two and three dimensional space in details. The topological inner structure of the phase singularities are obtained, and the topological charge of the phase singularities are expressed by the topological numbers: Hopf indices and Brouwer degrees. Furthermore, the topological invariant of the closed and knotted phase singularities in three dimensional space are also discussed in details.