# Spectral flow for Dirac operators with magnetic links

## TL;DR

This paper investigates the spectral properties of Dirac operators on the three-sphere with magnetic fields supported on links, revealing a periodicity in flux and deriving explicit spectral flow formulas related to link topology.

## Contribution

It introduces a detailed analysis of spectral flow for Dirac operators with magnetic links, including explicit formulas for unknots based on linking numbers and writhes.

## Key findings

- Spectral flow exhibits a $2\pi$-periodicity in flux for magnetic links.
- Explicit spectral flow formulas are derived for unknots using linking numbers and writhes.
- Spectral flow can be non-trivial for loops in the flux torus.

## Abstract

This paper is devoted to the study of the spectral properties of Dirac operators on the three-sphere with singular magnetic fields supported on smooth, oriented links. As for Aharonov-Bohm solenoids in Euclidean three-space, the flux carried by an oriented knot features a $2\pi$-periodicity of the associated operator. For a given link one thus obtains a family of Dirac operators indexed by a torus of fluxes. We study the spectral flow of paths of such operators corresponding to loops in this torus. The spectral flow is in general non-trivial. In the special case of a link of unknots we derive an explicit formula for the spectral flow of any loop on the torus of fluxes. It is given in terms of the linking numbers of the knots and their writhes.