Conneted sum of representations of knot groups

TL;DR

This paper introduces a new way to combine boundary-parabolic representations of knot groups called the connected sum, revealing properties like unique factorization and relationships between complex volume and Alexander polynomial.

Contribution

It defines the connected sum of representations, proves its key properties, and relates the complex volume and Alexander polynomial of the sum to those of the original representations.

Findings

Connected sum has the unique factorization property.

Complex volume of the sum equals the sum of individual volumes modulo iπ^2.

Twisted Alexander polynomial of the sum is the product of individual polynomials.

Abstract

When two boundary-parabolic representations of knot groups are given, we introduce the connected sum of these representations and show several natural properties including the unique factorization property. Furthermore, the complex volume of the connected sum is the sum of each complex volumes modulo $i\pi^2$ and the twisted Alexander polynomial of the connected sum is the product of each polynomials with normalization.