Gr(?)bner-Shirshov Basis For Kauffman Algebra

The Gr(?)bner-Shirshov basis theory is a new branch of algebra which was developed in the 1960s and 1970s by Buchberger(for commutative algebra),Bergman(for associative algebra)and Shirshov(for Lie algebra)in order to solve the reduction problem in alge-bra.The core result in the Gr(?)bner-Shirshov basis theory is the so called Composition-Diamond Lemma,which whenever we know the Gr(?)bner-Shirshov basis of an algebra,this lemma enables us to give a linear basis of the algebra.The Kauffman algebra is an important algebra which was developed independently by Temperly,Lieb and Kauffman in order to solve combinatorial problems in graph theory.In this thesis,we first give a Gr(?)bner-Shirshov basis for the Kauffman algebra by using the Shirshov algorithm and then,as an application,we give a linear basis for the Kauffman algebra by using the Composition-Diamond Lemma.The interesting point is that each of the elements in this basis is just in the Jones normal form.