Embedded graph invariants in Chern-Simons theory

TL;DR

This paper develops a method within Chern-Simons theory to define and compute embedded graph invariants, extending knot invariants to networks with arbitrary valence, and explores framing ambiguities.

Contribution

It introduces a generalized variational approach to define embedded graph invariants in Chern-Simons theory, including higher-order results and gauge-invariant operators.

Findings

Derived lowest-order invariants for arbitrary valence graphs

Introduced gauge-invariant operators for network analysis

Identified framing ambiguities related to vertex decomposition

Abstract

Chern-Simons gauge theory, since its inception as a topological quantum field theory, has proved to be a rich source of understanding for knot invariants. In this work the theory is used to explore the definition of the expectation value of a network of Wilson lines - an embedded graph invariant. Using a slight generalization of the variational method, lowest-order results for invariants for arbitrary valence graphs are derived; gauge invariant operators are introduced; and some higher order results are found. The method used here provides a Vassiliev-type definition of graph invariants which depend on both the embedding of the graph and the group structure of the gauge theory. It is found that one need not frame individual vertices. Though, without a global projection of the graph, there is an ambiguity in the relation of the decomposition of distinct vertices. It is suggested that framing may be seen as arising from this ambiguity - as a way of relating frames at distinct vertices.