How the Jones Polynomial Gives Rise to Physical States of Quantum General Relativity

TL;DR

This paper demonstrates that the Jones polynomial, a knot invariant, can generate an infinite set of solutions to all constraints in quantum gravity, revealing a profound link between knot theory and quantum gravitational states.

Contribution

It establishes a novel connection between the Jones polynomial and the solutions to quantum gravity constraints, advancing understanding of the quantum gravity solution space.

Findings

Jones polynomial yields solutions to all quantum gravity constraints

Deepens the link between knot invariants and quantum gravity

Suggests a dynamical relationship between knot theory and quantum states

Abstract

Solutions to both the diffeomorphism and the hamiltonian constraint of quantum gravity have been found in the loop representation, which is based on Ashtekar's new variables. While the diffeomorphism constraint is easily solved by considering loop functionals which are knot invariants, there remains the puzzle why several of the known knot invariants are also solutions to the hamiltonian constraint. We show how the Jones polynomial gives rise to an infinite set of solutions to all the constraints of quantum gravity thereby illuminating the structure of the space of solutions and suggesting the existance of a deep connection between quantum gravity and knot theory at a dynamical level.