Quantum knots and mosaics

Abstract

In this paper, we give a precise and workable definition of a knot system, the states of which are called knots. This definition can be viewed as a blueprint for the construction of an actual physical system. Moreover, this definition of a knot system is intended to represent the quantum embodiment of a closed knotted physical piece of rope. A knot, as a state of this system, represents the state of such a knotted closed piece of rope, i.e., the particular spatial configuration of the knot tied in the rope. Associated with a knot system is a group of unitary transformations, called the ambient group, which represents all possible ways of moving the rope around (without cutting the rope, and without letting the rope pass through itself.) Of course, unlike a classical closed piece of rope, a knot can exhibit non-classical behavior, such as superposition and entanglement. This raises some interesting and puzzling questions about the relation between topological and entanglement. The knot type of a knot is simply the orbit of the knot under the action of the ambient group. We investigate observables which are invariants of knot type. We also study the Hamiltonians associated with the generators of the ambient group, and briefly look at the tunneling of overcrossings into undercrossings. A basic building block in this paper is a mosaic system which is a formal (rewriting) system of symbol strings. We conjecture that this formal system fully captures in an axiomatic way all of the properties of tame knot theory.