# Partially abelian representations of knot groups

## TL;DR

This paper explores how solutions to Thurston's gluing equations with pinched octahedra relate to modifications in knot diagrams, providing insights into the structure of knot groups and their representations.

## Contribution

It introduces a new perspective on boundary parabolic solutions with pinched octahedra and their relation to knot modifications and holonomy representations.

## Key findings

- Pinched octahedra induce solutions for modified knots.
- Connections between pinched solutions and knot diagram changes.
- Examples including connected sum knots analyzed.

## Abstract

A knot complement admits a pseudo-hyperbolic structure by solving Thurston's gluing equations for an octahedral decomposition. It is known that a solution to these equations can be described in terms of region variables, also called $w$-variables. In this paper, we consider the case when pinched octahedra appear as a boundary parabolic solution in the decomposition. A $w$-solution with pinched octahedra induces a solution for a new knot obtained by changing the crossing or inserting a tangle at the pinched place. We discuss this phenomenon with corresponding holonomy representations and give some examples including ones obtained from connected sum.