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Quantum Knots and Lattices, or How Wiggle, Wag, and Tug Go Quantum
by
Samuel J. Lomonaco
University of Maryland Baltimore County
In the paper "Quantum Knots and Mosaics," QIP, (2008), the definition of quantum knots was based on the planar projections of knots (i.e., on knot diagrams) and the Reidemeister moves on these projections. In this paper, we take a different tack by creating a definition of quantum knots based on the cubic honeycomb decomposition of 3-space R³ (i.e., the cubic tessellation L_{ℓ} of R³ consisting of 2^{-ℓ}×2^{-ℓ}×2^{-ℓ} cubes) and a new set of knot moves, called wiggle, wag, and tug, which unlike the two dimensional Reidemeister moves are truly three dimensional moves. These new moves have been so named because they mimic how a dog might wag its tail.
We believe that these two different approaches to defining quantum knots are essentially equivalent, but that the above three dimensional moves have a definite advantage when it comes to the applications of knot theory to physics. More specifically, we contend that the new moves wiggle, wag, and tug are more "physics-friendly" than the Reidemeister ones. For unlike the Reidemeister moves, the new moves are three dimensional moves that respect the differential geometry of 3-space, which is indeed an essential component of physics. And moreover, unlike the Reidemeister moves, they can be transformed into infinitesimal moves and differential forms (which can in turn be integrated), which structures can be seamlessly interwoven with the equations of physics.
Our basic building block for constructing a quantum knot is a lattice knot, which is a knot in 3-space constructed from the edges of the cubic honeycomb L_{ℓ}. We then create a Hilbert space by identifying each edge of a bounded n×n×n region of the cubic honeycomb with a qubit. Lattice knots within this region then form the basis of a sub-Hilbert space K^{(ℓ,n)}. The states of K^{(ℓ,n)} are called quantum knots. The knot moves, wiggle, wag, and tug, are then naturally identified with the generators of a unitary group Λ_{ℓ,n}, called the lattice ambient group, acting on the Hilbert space K^{(ℓ,n)}.
This definition of a quantum knot can be viewed as a blueprint for the construction of an actual physical quantum system that represents the "quantum embodiment" of a closed knotted physical piece of rope. A quantum knot, as a state of this quantum knot system, represents the state of such a knotted closed piece of rope, i.e., the particular spacial configuration of the knot tied in the rope. The lattice ambient group Λ_{ℓ,n} 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.
After defining quantum knot type, we investigate quantum observables which are invariants of quantum knot type. Moreover, we also study the Hamiltonians associated with the generators of the lattice ambient group.
This talk is based on the joint paper Quantum Knots and Lattices, or How Wiggle, Wag, and Tug Go Quantum by Lomonaco and Kauffman, (to appear on quant-ph.)
Date received: February 23, 2009
Copyright © 2009 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # cayk-10.