Journal of Knot Theory and Its Ramifications

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 Appl139(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.

As a supplement to the paper [Prime knots with arc index up to 11 and an upper bound of arc index for non-alternating knots, J. Knot Theory Ramifications19(12) (2010) 1655–1672], we present minimal arc presentations of the prime knots up to arc index 11.