We show that the isomorphism problems for left distributive algebras, racks, quandles, and keis are as complex as possible in the sense of Borel reducibility. These various kinds of algebraic structure are important for their connections with the theory of knots, links and braids, and in particular, Joyce showed that quandles could be used as complete invariants for tame knots. However, quandles have heuristically seemed to be unsatisfactory invariants. Our result confirms this view, showing that from a set-theoretic perspective, classifying tame knots by quandles replaces one problem with (a special case of) a harder problem.