# Particle on a Torus Knot: Anholonomy and Hannay Angle

## TL;DR

This paper investigates the anholonomy and Hannay angle for particles moving along noncontractible loops on a torus, revealing unique behaviors and cancellations not seen in conventional closed-path analyses on spherical surfaces.

## Contribution

It extends the study of anholonomy and Hannay angle to noncontractible loops on a torus, highlighting novel effects for poloidal and toroidal paths.

## Key findings

- Anholonomy cancels out over some nontrivial cycles.
- Hannay angle is calculated for particles on revolving tori.
- Unique behaviors observed for poloidal paths.

## Abstract

The phenomenon of rotation of a vector under parallel transport along a closed path is known as anholonomy. In this paper we have studied the anholonomy for noncontractible loops - closed paths in a curved surface that do not enclose any area and hence Stokes theorem is not directly applicable. Examples of such closed paths are poloidal and toroidal loops and knots on a torus. The present study is distinct from conventional results on anholonomy for closed paths on $S_2$ since in the latter case all closed paths are contractible or trivial cycles. We find that for some nontrivial cycles the anholonomy cancels out over the complete cycle. Next we calculate Hannay angle for a particle traversing such noncontractible loops when the torus itself is revolving. Some new and interesting results are obtained especially for poloidal paths that is for paths that encircle the torus ring.