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

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.