An analytical expression for the third coefficient of the Jones Polynomial

TL;DR

This paper derives an explicit formula for the third coefficient of the Jones Polynomial in variable q, with applications discussed in the context of Quantum Gravity.

Contribution

It provides the first analytical expression for the third coefficient of the Jones Polynomial, enhancing understanding in knot theory and quantum gravity applications.

Findings

Derived an explicit formula for the third coefficient of the Jones Polynomial.

Applied the formula to problems in Quantum Gravity.

Facilitated further analytical studies of knot invariants in physics.

Abstract

An analytical expression for the third coefficient of the Jones Polynomial $P_J[\gamma,\, {\em e}^q]$ in the variable $q$ is reported. Applications of the result in Quantum Gravity are considered.