Knot groups, quandle extensions and orderability

Abstract: This paper gives a new way of characterizing L-space $3$-manifolds by using orderability of quandles. Hence, this answers a question of Adam Clay et al. [Question 1.1 of Canad. Math. Bull. 59 (2016), no. 3, 472-482]. We also investigate both the orderability and circular orderability of dynamical extensions of orderable quandles. We give conditions under which the conjugation quandle on a group, as an extension of the conjugation of a bi-orderable group by the conjugation of a right orderable group, is right orderable. We also study the right circular orderability of link quandles. We prove that the $n$-quandle $Q_n(L)$ of the link quandle of $L$ is not right circularly orderable and hence it is not right orderable. But on the other hand, we show that there are infinitely many links for which the $p$-enveloping group of the link quandle is right circularly orderable for any prime integer $p$.