  Using a model of idealized, crossed one-dimensional quantum wires we
construct a novel model for a single electron on tunneling-coupled systems of
one-dimensional quantum rings. We explore and find that topology can affect the
energetics of the system, and can introduce frustration in the three ring case.
We also study the special cases of an external magnetic field that controls the
complex phase of the tunneling matrix element, and introduce the knot theory
concept of "writhe" as a new topological quantity for distinguishing one hole
systems. We find that writhe not only determines the energetic response of the
system to magnetic field strength, but is also responsible for a single
particle topological quantum phase transition involving the ground state wave
function winding number.
