The Kauffman bracket and the Jones polynomial in quantum gravity

TL;DR

This paper explores how the Kauffman bracket and Jones polynomial relate to the Hamiltonian constraint in quantum gravity, showing the Kauffman bracket as a formal solution with a cosmological constant in an extended loop framework.

Contribution

It demonstrates that the Kauffman bracket solves the Hamiltonian constraint with cosmological constant to third order, using the extended loop representation and knot invariants from Chern-Simons theory.

Findings

Kauffman bracket is a formal solution with cosmological constant

Analysis performed in extended loop representation

Implications discussed for conventional loop representation

Abstract

An analysis of the action of the Hamiltonian constraint of quantum gravity on the Kauffman bracket and Jones knot polynomials is proposed. It is explicitely shown that the Kauffman bracket is a formal solution of the Hamiltonian constraint with cosmological constant ($\Lambda$) to third order in $\Lambda$. The calculation is performed in the extended loop representation of quantum gravity. The analysis makes use of the analytical expressions of the knot invariants in terms of the two and three point propagators of the Chern-Simons theory. Some particularities of the extended loop calculus are considered and the implications of the results to the case of the conventional loop representation are discussed.