Knotted 4-regular graphs II: Consistent application of the Pachner moves

TL;DR

This paper explores the consistent application of Pachner moves to knotted 4-regular graphs in loop quantum gravity, emphasizing the importance of framing and twists, and introduces algebraic tools for generalized braid representations.

Contribution

It demonstrates how to apply Pachner moves consistently on framed knotted graphs with twists and introduces an algebraic framework for generalized braid group representations.

Findings

Pachner moves require specific twist criteria for consistency.

Framed graphs with twists can be manipulated using Pachner moves.

An algebraic object enables representation of knotted graphs in a generalized braid group.

Abstract

A common choice for the evolution of the knotted graphs in loop quantum gravity is to use the Pachner moves, adapted to graphs from their dual triangulations. Here, we show that the natural way to consistently use these moves is on framed graphs with edge twists, where the Pachner moves can only be performed when the twists, and the vertices the edges are incident on, meet certain criteria. For other twists, one can introduce an algebraic object, which allow any knotted graph with framed edges to be written in terms of a generalized braid group.