On the quandles of isometries of the hyperbolic 3-space

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 PSL (2, C) of hyperbolic 3-space and its subquandles. We introduce a quandle, denoted by Q (, ), associated with a pair (, ). Here, is a Kleinian group, and is a non-trivial element of. This construction can be regarded as a generalization of knot quandles to hyperbolic knots. Moreover, for pairs (, ) satisfying certain conditions, we construct the canonical map from Q (, ) to the conjugate quandle of PSL (2, C), which is an injective quandle homomorphism with a discrete image.