Finite involutory quandles of two-bridge links with an axis

To better understand the fundamental quandle of a knot or link, it can be useful to look at finite quotients of the quandle. One such quotient is the n-quandle (or, when n = 2, the involutory quandle). Hoste and Shanahan [J. Hoste and P. D. Shanahan, Links with finite n-quandles, Algebraic Geom. Topol. 17 (2017) 2807-2823.] gave a complete list of the links which have finite n-quandles; it remained to give explicit descriptions of these quandles. This has been done for several cases in [A. Crans, J. Hoste, B. Mellor and P. D. Shanahan, Finite n-qundles of torus and two-bridge links, J. Knot Theory Ramifications 28 (2019) 1950028; J. Hoste and P. D. Shanahan, Involutory quandles of (2, 2, r)-Montesinos links, J. Knot Theory Ramifications 26 (2017)]; in this work, we continue this project and explicitly describe the Cayley graphs for the finite involutory quandles of two-bridge links with an axis.