Burnside kei

Abstract

This paper is motivated by a general question: for which values of k and n is the universal Burnside kei of k generators and Kei "exponent" n, $\bar{Q}(k,n)$, finite? It is known (starting from the work of M. Takasaki (1942)) that $\bar{Q}(2,n)$ is isomorphic to the dihedral quandle Z_n and $\bar{Q}(3,3)$ is isomorphic to Z_3 + Z_3. In this paper we give descriptions of $\bar{Q}(4,3)$ and $\bar{Q}(3,4)$. We also investigate some properties of arbitrary quandles satisfying the universal Burnside relation (of "exponent" n) a=...a*b*...*a*b. In particular, we prove that the order of a finite commutative kei is a power of 3. Invariants of links related to Burnside kei $\bar{Q}(k,n)$ are invariant under n-moves.

15 pages, 4 figures