Quantum Knots and Lattices, or a Blueprint for Quantum Systems that Do Rope Tricks

TL;DR

This paper introduces a quantum system modeling knotted ropes using a lattice framework, replacing classical knot moves with physics-friendly operations, and explores its mathematical properties and potential physical implementation.

Contribution

It defines a novel quantum knot system based on lattice moves that respect differential geometry, providing a new approach to quantum topology and potential physical realizability.

Findings

Defines quantum knots using lattice moves wiggle, wag, and tug.

Introduces the lattice ambient group acting on the quantum knot states.

Analyzes knot invariants and Hamiltonians within the quantum knot framework.

Abstract

Using the cubic honeycomb (cubic tessellation) of Euclidean 3-space, we define a quantum system whose states, called quantum knots, represent a closed knotted piece of rope, i.e., represent the particular spatial configuration of a knot tied in a rope in 3-space. This quantum system, called a quantum knot system, is physically implementable in the same sense as Shor's quantum factoring algorithm is implementable. To define a quantum knot system, we replace the standard three Reidemeister knot moves with an equivalent set of three moves, called respectively wiggle, wag, and tug, so named because they mimic how a dog might wag its tail. We argue that these moves are in fact more "physics friendly" because, unlike the Reidemeister moves, they respect the differential geometry of 3-space, and moreover they can be transformed into infinitesimal moves. These three moves wiggle, wag, and tug generate a unitary group, called the lattice ambient group, which acts on the state space of the quantum system. The lattice ambient group represents all possible ways of moving a rope around in 3-space without cutting the rope, and without letting the rope pass through itself. We then investigate those quantum observables of the quantum knot system which are knot invariants. We also study Hamiltonians associated with the generators of the lattice ambient group. We conclude with a list of open questions.