ABSTRACT
This chapter describes the universal knot invariant discovered independently by S. V. Matveev and D. Joyce. In Matveev's work and in other works by Russian authors, this invariant is usually called the distributive groupoid; in Western literature it is usually called quandle. The chapter provides some series of "weaker" invariants coming from the knot quandle; the series of invariants to be constructed are easier to calculate and to compare. The quandle is a complete knot invariant because it contains the information about the fundamental group and "a bit more". To prove this fact about the completeness of the quandle, the chapter utilizes one very strong result by F. Waldhausen concerning three-dimensional topological surgery. The universal link invariant corresponding to the universal algebra is the strongest one. However, it has a significant disadvantage: it is difficult to recognise two different presentations of an element in the algebra.
