# Helicity Conservation, and Twisted Seifert Surfaces for Superfluid Vortices

## TL;DR

This paper derives the contributions to helicity in superfluid vortices, revealing that the continuum helicity is always zero and proposing the Gauss-linking number as a more suitable measure, with implications for connecting microscopic and macroscopic descriptions.

## Contribution

It introduces a first-principles derivation of helicity contributions in superfluids and highlights the importance of the Gauss-linking number over continuum helicity for superfluid vortices.

## Key findings

- Continuum helicity for superfluids is always zero due to Seifert framing.
- The Gauss-linking number better captures vortex linking in superfluids.
- A quasiclassical limit connects microscopic and macroscopic superfluid descriptions.

## Abstract

Starting from the continuum definition of helicity, we derive from first principles its different contributions for superfluid vortices. Our analysis shows that an internal twist contribution emerges naturally from the mathematical derivation. This reveals that the spanwise vector that is used to characterise the twist contribution must point in the direction of a surface of constant velocity potential. An immediate consequence of the Seifert framing is that the continuum definition of helicity for a superfluid is trivially zero at all times. It follows that the Gauss-linking number is a more appropriate definition of helicity for superfluids. Despite this, we explain how a quasiclassical limit can arise in a superfluid in which the continuum definition for helicity can be used. This provides a clear connection between a microscopic and a macroscopic description of a superfluid as provided by the Hall-Vinen-Bekarevich-Khalatnikov equations. This leads to consistency with the definition of helicity used for classical vortices.