On the quandles of isometries of the hyperbolic 3-space

TL;DR

This paper explores the algebraic structure called quandles related to hyperbolic 3-space isometries, generalizing knot quandles to hyperbolic knots and establishing a canonical injective homomorphism for certain pairs.

Contribution

It introduces a new class of quandles associated with Kleinian groups and elements, extending knot quandle concepts to hyperbolic geometry and constructing canonical maps with specific properties.

Findings

Defined a quandle $Q(\Gamma, \gamma)$ for Kleinian groups and elements.

Constructed an injective quandle homomorphism to the conjugate quandle of $ ext{PSL}(2,\mathbb{C})$.

Generalized knot quandles to hyperbolic knots.

Abstract

A quandle is an algebraic structure whose axioms are related to the Reidemeister moves used in knot theory. In this paper, we investigate the conjugate quandle of the orientation-preserving isometry group $\mathrm{PSL}(2, \mathbb{C})$ of hyperbolic 3-space and its subquandles. We introduce a quandle, denoted by $Q(\Gamma, \gamma)$, associated with a pair $(\Gamma, \gamma)$. Here, $\Gamma$ is a Kleinian group, and $\gamma$ is a non-trivial element of $\Gamma$. This construction can be regarded as a generalization of knot quandles to hyperbolic knots. Moreover, for pairs $(\Gamma, \gamma)$ satisfying certain conditions, we construct the canonical map from $Q(\Gamma, \gamma)$ to the conjugate quandle of $\mathrm{PSL}(2, \mathbb{C})$, which is an injective quandle homomorphism with a discrete image.