A Diagrammatic method for an invariant related to the Chern-Simons perturbation theory

TL;DR

This paper introduces a diagrammatic approach to compute the defect invariant of 3-manifolds with knot exteriors, utilizing knot diagrams to simplify calculations related to Chern-Simons perturbation theory.

Contribution

It presents a novel diagrammatic method for evaluating the defect invariant associated with Chern-Simons perturbation theory for knot exteriors.

Findings

Provides a practical diagrammatic technique for computing the invariant

Simplifies calculations of the defect for knot exteriors

Enhances understanding of invariants in 3-manifold topology

Abstract

The defect $d(M,\rho)$ is an invariant of a compact oriented 3-manifold $M$ with a representation $\rho$ of the fundamental group. In this article we give a diagrammatic method for $d$ of knot exteriors by using knot diagrams.