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 Clay et al. (Question 1.1 of Can Math Bull 59(3):472–482, 2016). 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 a link L in the 3-sphere 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.