On the quandles of isometries of the hyperbolic 3-space

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