An Exact Expression for a Flat Connection on the Complement of a Torus Knot

TL;DR

This paper derives an exact mathematical expression for a flat connection on the complement of a torus knot, advancing understanding in knot theory and potential physical phenomena like the Aharonov-Bohm effect.

Contribution

It provides a novel exact formula for flat connections on torus knot complements, aiding in knot group representations and topological invariants.

Findings

Exact expression for flat connection derived

Advances in knot group representation construction

Potential implications for generalized Aharonov-Bohm effect

Abstract

Simple physics ideas are used to derive an exact expression for a flat connection on the complement of a torus knot. The result is of some mathematical importance in the context of constructing representations of the knot group -- a topological invariant of the knot. It is also a step forward in the direction of obtaining a generalisation of the Aharonov-Bohm effect, in which charged particles moving through force-free regions are scattered by impenetrable, knotted solenoids.