Identities of the Kauffman Monoid ${\mathcal{K}}_4$ and of the Jones monoid ${\mathcal{J}}_4$

Abstract

Kauffman monoids ${\mathcal{K}}_n$ and Jones monoids ${\mathcal{J}}_n$, $n=2,3,\dots$, are two families of monoids relevant in knot theory. We prove a somewhat counterintuitive result that the Kauffman monoids ${\mathcal{K}}_3$ and ${\mathcal{K}}_4$ satisfy exactly the same identities. This leads to a polynomial time algorithm to check whether a given identity holds in ${\mathcal{K}}_4$. As a byproduct, we also find a polynomial time algorithm for checking identities in the Jones monoid ${\mathcal{J}}_4$.

25 pages, 7 figures