Burnside kei
Volume 190 / 2006
Fundamenta Mathematicae 190 (2006), 211-229
MSC: Primary 57M25; Secondary 55N99, 20D99.
DOI: 10.4064/fm190-0-8
Abstract
This paper is motivated by a general question: for which values of $k$ and $n$ is the universal Burnside kei $\overline{Q}{\partial} (k,n)$ finite? It is known (starting from the work of M. Takasaki (1942)) that $\overline{Q}{\partial}(2,n)$ is isomorphic to the dihedral quandle $Z_n$ and $\overline{Q}{\partial}(3,3)$ is isomorphic to $Z_3 \oplus Z_3$. In this paper, we give a description of the algebraic structure for Burnside keis $\overline{Q}{\partial}(4,3)$ and $\overline{Q}{\partial}(3,4)$. We also investigate some properties of arbitrary quandles satisfying the universal Burnside relation $a=\cdots a*b*\cdots *a*b$. Invariants of links related to the Burnside kei $\overline{Q}{\partial}(k,n)$ are invariant under $n$-moves.
