The concept of quandles was explicitly presented in the independent works of D. Joyce [J. Pure Appl. Algebra 23, 37–65 (1982; Zbl 0474.57003)] and S. V. Matveev [Math. USSR, Sb. 47, 73–83 (1984; Zbl 0523.57006); translation from Mat. Sb., Nov. Ser. 119(161), No. 1, 78–88 (1982)]. To each oriented diagram \(D_K\) of an oriented knot \(K\) in \(\mathbb{R}^3\) they associate the quandle \(Q(K)\) which does not change if we apply the Reidemeister moves to the diagram \(D_K\). Joyce and Matveev proved that two knot quandles \(Q(K_1)\) and \(Q(K_2)\) are isomorphic if and only if there is a homeomorphism (possibly reversing orientation) of the ambient space \(\mathbb{R}^3\) which maps \(K_1\) to \(K_2\). The knot quandle is a very strong invariant for knots in \(\mathbb{R}^3\), however, usually it is very difficult to determine if two knot quandles are isomorphic. In [A. D. Brooke-Taylor and S. K. Miller, J. Aust. Math. Soc. 108, No. 2, 262–277 (2020; Zbl 1482.20039)] it is shown that the isomorphism problem for quandles is, from the perspective of Borel reducibility, fundamentally difficult (Borel complete).
Sometimes homomorphisms from knot quandles to simpler quandles provide useful information that helps determine whether two knot quandles are isomorphic. One of the homomorphisms that provides useful information about the quandle \(Q(K)\) is the homomorphism \(Q(K)\to Q_n(K)\) which sends the quandle \(Q(K)\) to its quotient \(Q_n(K)\) by the relations \((\dots((x*y)*y)*\dots)*y\) for all \(x,y\in Q(K)\), where \(*\) is a quandle operation, and \(y\) appears \(n\) times in the formula. The authors of the paper under review study the quandle \(Q_2(L)\) in the case when \(L\) is a two-bridge link with an axis. In this case the authors calculate the order of \(Q_2(L)\) and give an explicit description of the Cayley graphs for these quandles.