Vassiliev Theory

There exists a natural filtration on the module freely generated by knots (or links). This filtration is called the Vassiliev filtration and has many nice properties. In particular every quotient of this filtration is finite dimensional. A knot invariant which vanishes on some module of this filtration is called a Vassiliev invariant. Almost every knot invariant defined in algebraic term can be describe in term of Vassiliev invariants. Unfortunately the structure of all such invariants is completely unknown. The Kontsevich integral constructed in Christine Lescop’s lecture [L] is, in some sense, the universal Vassiliev invariant. It takes values in a module A of 3-valent diagrams. So a good way to construct a knot invariant is to compose the Kontsevich integral with a linear homomorphism defined on A. Every Lie algebra equipped with a nonsingular bilinear symmetric invariant form produces a linear homomorphism on A and therefore a knot invariant. If the Lie algebra belongs to the A series, the induced knot invariant is the HOMFLY polynomial. If the Lie algebra belongs to the B-C-D series, one gets the Kauffman polynomial. The Kauffman bracket is obtained by the Lie algebra sl2. 1. VASSILIEV INVARIANTS 1.1 Knots and links invariants A link is a compact 1-dimensional smooth submanifold of R. A connected link is called a knot. A link may be oriented or not. Every knot is the image of a embedding f from the circle S into R. For a link, the situation is similar but the embedding is defined on a disjoint union of finitely many copies of the circle. 1.2 Definitions. A link L is called banded if L is equipped with a vector field V from L to R such that V (x) is transverse to L for every point x ∈ L. A link L is called framed if it is oriented and banded. Université Paris VII, UMR 7586, UFR de mathématiques, Case 7012, 2 place Jussieu 75251 Paris Cedex 05 – Email: [email protected]