On representations of 2-bridge knot groups in quaternion algebras

TL;DR

This paper investigates how 2-bridge knot groups can be represented within quaternion algebras, connecting algebraic structures to geometric actions in various 3-dimensional spaces.

Contribution

It introduces the variety of affine c-representations for 2-bridge knot groups and analyzes their algebraic and geometric properties using quaternion algebra theory.

Findings

Characterization of representations in quaternion algebra units

Description of the algebraic variety of affine c-representations

Insights into deformations of these representations

Abstract

Representations of two bridge knot groups in the isometry group of some complete Riemannian 3-manifolds as $E^{3}$ (Euclidean 3-space), $H^{3}$ (hyperbolic 3-space) and $ E^{2,1}$ (Minkowski 3-space), using quaternion algebra theory, are studied. We study the different representations of a 2-generator group in which the generators are send to conjugate elements, by analyzing the points of an algebraic variety, that we call the \emph{variety of affine c-representations of}$G$. Each point in this variety correspond to a representation in the unit group of a quaternion algebra and their affine deformations.