On a Background of the Existence of Multi-variable Link Invariants

Abstract

Abstract. We present a quantum theorical background of the existence of multi-variable link invariants, for example the Kauffman polynomial, by observing the quan-tum (sl(2,C),ad)-invariant from the Kontsevich invariant point of view. The backgroundimplies that the Kauffman polynomial can be studied by using the sl(N,C)-skein theorysimilar to the Jones polynomial and the HOMFLY polynomial.1. IntroductionIn 1980s–90s, many multi-variable link invariants had been successfully constructed,for example, the Λ-polynomial ([7]), the Q-polynomial ([1], [4]) and the Kauffman poly-nomial. Why was it possible? In this paper, we present a quantum theorical backgroundof the existence of the above multi-variable link invariants by observing the quantum(sl(2,C),ad)-invariant from the Kontsevich invariant point of view.According to [8], the quantum (so(N),ρ 0 )-invariant, where ρ 0 is the fundamentalrepresentation of so(N), is a specialization of the Kauffman polynomial F(L;a,z) in theLaurent polynomial ring Z[a,a