Quantum knots and quantum braids: Their possible application to superfluid vortices and to topological quantum computing in optical lattices.
In this talk, we show how to reconstruct knot theory in such a way that it is intimately related to quantum physics. In particular, we give a blueprint for creating a quantum system that has the dynamic behavior of a closed knotted piece of rope moving in 3-space. Within this framework, knot invariants become physically measurable quantum observables, knot moves become unitary transformations, with knot dynamics determined by Schroedinger’s equation. The same approach can also be applied to the theory of braids. Toward the end of the talk, we look at possible applications to superfluid vortices and to topological quantum computing in optical lattices.