# Vassiliev knot invariants

## Idea

The space of knots in $\mathbb{R}^3$ (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 discriminant 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.

###### Definition

A Vassiliev invariant of degree (or order) $\le n$ is a knot invariant whose extension to singular knots (with double points) vanishes on all singular knots with more than $n$ double points.

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.

## Properties

A function which is constant on nonsingular knots may be extended to a Vassiliev invariant of degree 0 by applying the Vassiliev skein relations, and conversely, any Vassiliev invariant of degree 0 must be constant on nonsingular knots. Likewise, any Vassiliev invariant of degree 1 must be constant on nonsingular knots.

## Chord diagrams and weight systems

Any singular knot $f : S^1 \to \mathbb{R}^3$ with $n$ distinct double points $x_1,\dots,x_n \in \mathbb{R}^3$ gives rise to a chord diagram of order $n$, consisting of the circle $S^1$ with a chord connecting each pair of points $f^{-1}(x_1), \dots, f^{-1}(x_n)$.

The importance of this construction for singular knots comes from the fact that any finite type invariant determines a function on chord diagrams:

###### Theorem

Let $v$ be a Vassiliev invariant of degree $\le n$. Then the value of $v$ on a singular knot with $n$ distinct double points depends only on the chord diagram of the knot, and not on the knot itself.

Conversely, one can ask which functions on chord diagrams come from finite type invariants. The answer is that Vassiliev invariants (of degree $\le n$) can essentially be identified with weight systems (of order $n$), which are functions on chord diagrams (of order $n$) satisfying two properties called the “1-term relation” (or “framing independence”) and the “4-term relation”: see Theorem 1 of Bar-Natan (or Theorem 6.2.13 of Lando & Zvonkin).

## Examples

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

## References

• V. A. Vassiliev, Complements of discriminants of smooth maps: topology and applications, Amer. Math. Soc. 1992.
• Dror Bar-Natan, On the Vassiliev knot invariants, Topology 34 (1995), 423-472. (html)
• There is a huge online bibliography on Vassiliev invariants maintained by Dror Bar-Natan and Sergei Duzhin here.
• S. Chmutov, S. Duzhin, J. Mostovoy, Introduction to Vassiliev knot invariants, arxiv/1103.5628
• Wolframworld Vassiliev invariant