Knotted 4-regular graphs: polynomial invariants and the Pachner moves

TL;DR

This paper develops polynomial invariants for knotted 4-regular graphs in loop quantum gravity, exploring their behavior under Pachner moves to distinguish quantum geometric states.

Contribution

Introduces two novel polynomial invariants for knotted 4-regular graphs and analyzes their transformation properties under Pachner moves.

Findings

Two polynomial invariants successfully characterize knotted 4-regular graphs.

The quandle-based invariant's behavior under Pachner moves is systematically studied.

Potential applications in classifying quantum geometric states in loop quantum gravity.

Abstract

In loop quantum gravity, states of quantum geometry are represented by classes of knotted graphs, equivalent under diffeomorphisms. Thus, it is worthwhile to enumerate and distinguish these classes. This paper looks at the case of 4-regular graphs, which have an interpretation as objects dual to triangulations of three-dimensional manifolds. Two different polynomial invariants are developed to characterize these graphs -- one inspired by the Kauffman bracket relations, and the other based on quandles. How the latter invariant changes under the Pachner moves acting on the graphs is then studied.