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Centre for Mathematical Physics and Stochastics
Department of Mathematical Sciences, University of Aarhus

Funded by The Danish National Research Foundation

MPS-LN 2000-6
February 2000

Vassiliev Theory


P. Vogel


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 terms can be described in terms of Vassiliev invariants. Unfortunately the structure of all such invariants is completely unknown. The Kontsevich integral 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 $sl_2$.

The structure of A is more or less unknown. This module is actually a polynomial algebra, but the number $d_n$ of generators in degree n is known only for n<13. A Lie algebra L induces an algebra homomorphism from the Hopf algebra A to the center of the enveloping algebra of L. So, in some sense, there a universal algebra L over an algebra $\Lambda$ such that A is the center of the enveloping algebra of L. This Lie algebra is defined as a category satisfying some conditions. There are many conjectures about this universal Lie algebra and the coefficient algebra $\Lambda$. The decomposition of $\L^{\otimes p}$ in simple modules is given for $p\leq 3$.

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