Kauffman Monoids

Abstract

This paper gives a self-contained and complete proof of the isomorphism of freely generated monoids extracted from Temperley-Lieb algebras with monoids made of Kauffman's diagrams.

Identities of the Kauffman monoid K3

“…for some ℓ ≥ 0 and some a 1 < · · · < a k , b 1 < · · · < b k , or of the form c ℓ for some ℓ ≥ 0. The proofs of the next statement can be found in [Borisavljević et al, 2002] and [Bokut' and Lee, 2005].…”

Ideal Structure of the Kauffman and Related Monoids

“…The Jones monoid J n : As noted in the introduction, [1] de…nitively established relations (1) as a presentation for K n : A presentation for J n is obtained by adjoining c = 1 to the relations (1) and is equivalent to the monoid presentation with generators h i for i = 1; : : : n 1 and relations…”

“…The basis elements of a Temperley-Lieb algebra constitute a monoid K n generated by c; h 1 ; : : : h n 1 and subject to the relations (1), so that the Temperley-Lieb algebra is the semigroup algebra C[K n ] of K n . A representation of the monoid K n by isotopy classes in point-set topology was found and popularised, perhaps most notably by Jones, in section 4 of [4], and by Kau¤man [5].…”

“…It was widely held that this representation was in fact an isomorphism, but no complete proof of this was published until Borisavljević, Došen and Petrić [1]. They suggested the name Kau¤ man monoid for K n to honour the contributions of L. H. Kau¤man to the theory of K n and C [K n ].…”

“…This is what we attempt here. Our approach uses the quotient semigroup J n = K n = (c; 1) ] obtained by setting c = 1 in relations (1), which we name here the Jones monoid ( with thanks to an anonymous referee for the suggestion. The corresponding algebra C [J n ] is a quotient algebra of T L n ; namely T L n = hc 1i.…”