Variational derivation of exact skein relations from Chern--Simons theories

TL;DR

This paper introduces a variational method to derive exact skein relations for Wilson loop invariants in Chern--Simons theories, extending to intersecting knots and links, with implications for quantum gravity states.

Contribution

It generalizes variational techniques to compute exact skein relations and extends knot invariants to intersecting knots, aligning with Wilson loop identities.

Findings

Derived exact skein relations for Wilson loops

Extended knot invariants to intersecting knots and links

Provided insights into non-planar intersections and kinks

Abstract

The expectation value of a Wilson loop in a Chern--Simons theory is a knot invariant. Its skein relations have been derived in a variety of ways, including variational methods in which small deformations of the loop are made and the changes evaluated. The latter method only allowed to obtain approximate expressions for the skein relations. We present a generalization of this idea that allows to compute the exact form of the skein relations. Moreover, it requires to generalize the resulting knot invariants to intersecting knots and links in a manner consistent with the Mandelstam identities satisfied by the Wilson loops. This allows for the first time to derive the full expression for knot invariants that are suitable candidates for quantum states of gravity (and supergravity) in the loop representation. The new approach leads to several new insights in intersecting knot theory, in particular the role of non-planar intersections and intersections with kinks.