Vassiliev Invariants

Idea

The space of knots in ${ℝ}^3$ (or $S^3$) is an open submanifold of the smooth loop space. Knot invariants are locally constant functions on this manifold. The complement of the space of knots is called the discriminent? and consists of all singular knots.

If we consider those singular knots with only a finite number of double points, we can build a cubical complex? from this data. The vertices in the complex are labelled by the isotopy classes of knots, and more generally the $n$-cubes by the isotopy classes of singular knots with $n$ double points (and a few other technical pieces of information). The boundary operator resolves a double crossing either upwards or downwards according to the orientation at the crossing.

A Vassiliev invariant is simply a cubical morphism from this complex to an abelian group that vanishes above a certain degree.

Definition

One does not need the language of cubical complexes to define Vassiliev invariants. Rather, there is a general method whereby a knot invariant can be extended to all singular knots with only finitely many double points (and no other singularities) using the Vassiliev skein relations.

As is standard, it is of degree $n$ if it is of degree $\le n$ but not $\le n-1$. Vassiliev invariants are also called finite type invariants.

Remarks

The degree of Vassiliev invariants defines a filtration on the space of knots (and more particularly, on the algebra of knots?). Two knots are $n$-equivalent if all the Vassiliev invariants of degree $\le n$ agree on them. In particular, a knot that is $n$-equivalent to the unknot is said to be $n$-trivial.

Examples

1. The $n$th coefficient of the Conway polynomial is a Vassiliev invariant of order $\le n$.

References

• There is a huge online bibliography on Vassiliev invariants maintained by Dror Bar-Natan and Sergei Duzhin here.
• Chmutov, S., Duzhin, S., & Mostovoy, J. Introduction to Vassiliev Knot Invariants. http://arxiv.org/abs/1103.5628