Quantum Knots Revisited

Samuel J. Lomonaco

Samuel J. Lomonaco, Louis Hirsch Kauffman

Samuel J. Lomonaco, Louis Hirsch Kauffman

In this paper, we give a precise and workable definition of a quantum knot system, the states of which are called quantum knots. This definition can be viewed as a blueprint for the construction of an actual physical quantum system. Moreover, this definition of a quantum knot system is intended to represent the "quantum embodiment" of a closed knotted physical piece of rope. A quantum 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 quantum 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 quantum knot can exhibit non-classical behavior, such as quantum superposition and quantum entanglement. This raises some interesting and puzzling questions about the relation between topological and quantum entanglement. The knot type of a quantum knot is simply the orbit of the quantum knot under the action of the ambient group. We investigate quantum observables which are invariants of quantum knot type. We also study the Hamiltonians associated with the generators of the ambient group, and briefly look at the quantum 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.

David S. Hall, Michael Ray, Konstantin Tiurev +3 more

Graham P. Collins

S. Schaveling

In my research, we developed a method to distinguish knots. A knot is a mathematical depiction of the everyday knot that occurs in ropes and cords. The distinguishing of knots is a computationally hard problem, which is equivalent with the untangling of knots. Often, knots are distinguished by a so called knot invariant. Such an invariant is a label that can be attributed to any knot, such that the label is identical for two identical knots. The theory of knots is closely related to quantum computing and lattices of atoms. The distinguishing of knots is a problem that can be solved in an exponential number of computations, in the number of crossings of a knot. Our invariants can be computed in polynomial time. This means that the number of computations is a power of the number of crossings of a knot. Another advantage of our method is that for the knots that have been calculated, the invariant was able to distinguish as the most effective methods combined. In fact, by removing unnecessary information from the conventional knot invariants, we obtained an invariant that can be computed much faster, and is as effective as the usual invariants.