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 [16]. The theoretical importance of rack cohomology is exemplified by a theorem [13] identifying the homotopy groups of a rack space (see §3 ) 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.
In [6] an attempt was made to classify quandles of small order. There appear to be 404 quandles of order 5, 6658 quandles of order 6, 152900 quandles of order 7 and 5225916 quandles of order 8. For order 9 the search space was too large to finish the computation. If however one restricts to the important subclass of connected quandles then classification seems to be more accessible to computation, comparable to the classification of groups of given order. It is the purpose of this note to classify connected quandles up to order 14, and in particular to show that there is no connected quandle of order 14.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.