Is the third coefficient of the Jones knot polynomial a quantum state of gravity?

TL;DR

This paper investigates whether the third coefficient of the Jones knot polynomial can serve as a quantum state of gravity, finding it does not for general loops but may have potential for ordinary loops.

Contribution

It analyzes the conjecture that Jones polynomial coefficients are solutions to quantum gravity constraints, specifically at third order, using the extended loop representation.

Findings

Hamiltonian does not annihilate $J_3$ for general extended loops

For ordinary loops, $J_3$ has a meaningful geometric interpretation

Potential for $J_3$ to represent quantum gravity states in specific cases

Abstract

Some time ago it was conjectured that the coefficients of an expansion of the Jones polynomial in terms of the cosmological constant could provide an infinite string of knot invariants that are solutions of the vacuum Hamiltonian constraint of quantum gravity in the loop representation. Here we discuss the status of this conjecture at third order in the cosmological constant. The calculation is performed in the extended loop representation, a generalization of the loop representation. It is shown that the the Hamiltonian does not annihilate the third coefficient of the Jones polynomal ($J_3$) for general extended loops. For ordinary loops the result acquires an interesting geometrical meaning and new possibilities appear for $J_3$ to represent a quantum state of gravity.