On Kauffman's Knot Invariants Arising from Finite-Dimensional Hopf Algebras
David E. Radford
Abstract
This chapter discusses the knot invariants described by Kauffman which arise from finite-dimensional ribbon Hopf algebras over a field k and describes the algebra involved in their construction and computation. It also constructs an extensive family of finite-dimensional ribbon Hopf algebras for generating invariants. The chapter discusses briefly some aspects of the theory of finite-dimensional Hopf algebras which play an important role in the sequel. It examines the connection between grouplike elements and integrals. The chapter reviews the notions of finite-dimensional quasitriangular, ribbon and factorizable Hopf algebras and explore connections between them. It introduces the notion of quasitriangular envelope and shows that the Drinfel'd double of a finitedimensional Hopf algebra has a universal description in this context. It is a pleasure to acknowledge the many conversations with Louis Kauffman concerning the topology and algebra of knots, links and 3-manifolds.