THE ALGEBRA OF RACK AND QUANDLE COHOMOLOGY

This paper presents the first complete calculation of the cohomology of any nontrivial quandle, establishing that this cohomology exhibits a very rich and interesting algebraic structure. Rack and quandle cohomology have been applied in recent years to attack a number of problems in the theory of knots and their generalizations like virtual knots and higher-dimensional knots. An example of this is estimating the minimal number of triple points of surface knots [E. Hatakenaka, An estimate of the triple point numbers of surface knots by quandle cocycle invariants, Topology Appl 139(1-3) (2004) 129-144.]. The theoretical importance of rack cohomology is exemplified by a theorem [R. Fenn, C. Rourke and B. Sanderson, James bundles and applications, Proc. London Math. Soc. (3) 89(1) (2004) 217-240] identifying the homotopy groups of a rack space with a group of bordism classes of high-dimensional knots. There are also relations with other fields, like the study of solutions of the Yang-Baxter equations.