Identities of the kauffman monoid K4 and of the Jones Monoid J4

Abstract

Kauffman monoids Kn and Jones monoids Jn, n=2,3,…, are two families of monoids relevant in knot theory. We prove a somewhat counterintuitive result that the Kauffman monoids K3 and K4 satisfy exactly the same identities. This leads to a polynomial time algorithm to check whether a given identity holds in K4. As a byproduct, we also find a polynomial time algorithm for checking identities in the Jones monoid J4.