ABSTRACT
This chapter discusses the Casson invariant of a closed three-manifold for a knot in a closed three-manifold. This is a chord diagram version of the method noted in the end of by Lickorish. It also discusses the extension of the construction of the Casson invariant from the universal Vassiliev-Kontsevich invariant by adding the 3T relation and some natural relations. The semi-simple quotient of this representation is isomorphic to a disjoint union of two copies of the natural representation of SL(2, Z). The contribution from the extension of SL(2, Z) appears in the Levi part of this representation, and this part seems to contain information for the Casson invariant. The Kirby moves correspond to Kirby’s handle slide move, where any component can be changed only along a thin line component since the thick line component is a knot and is not applied the surgery.
