Vassiliev Invariant

A Vassiliev invariant, also called a finite type invariant, is a knot invariant whose extension to singular knots vanishes whenever the number of ordinary double points exceeds some fixed nonnegative integer. The notion was independently introduced by V. Vassiliev and M. Goussarov around 1989 (Duzhin 1999). Vassiliev's approach is based on the study of discriminants in the (infinite-dimensional) spaces of smooth maps from one manifold into another. By definition, the discriminant consists of all maps with singularities.

For example, consider the space of all smooth maps from the circle into three-dimensional Euclidean space . If is an embedding (i.e., has no singular points), then it represents a knot. The complement of the set of all knots is the discriminant . It consists of all smooth maps from into that have singularities, either local, where , or nonlocal, where is not injective. Two knots are equivalent iff they can be joined by a path in the space that does not intersect the discriminant. Therefore, knot types are in one-to-one correspondence with the connected components of the complement , and knot invariants with values in an Abelian group are nothing but cohomology classes from . The filtration of by subspaces corresponding to singular knots with a given number of ordinary double points gives rise to a spectral sequence, which contains, in particular, the spaces of finite type invariants.

Birman and Lin (1993) have contributed significantly to the simplification of Vassiliev's original techniques. In particular, they explained the relation between Jones polynomials and finite type invariants (Peterson 1992, Birman and Lin 1993, Bar-Natan 1995) and emphasized the role of the algebra of chord diagrams. In fact, substituting the power series for as the variable in the Jones polynomial yields a power series whose coefficients are Vassiliev invariants (Birman and Lin 1993). Kontsevich (1993) proved the first difficult theorem about Vassiliev invariants with the help of the Kontsevich integral. Bar-Natan undertook a thorough study of Vassiliev invariants; in particular, he showed the importance of the algebra of Feynman diagrams and diagrams with uni- and tri-valent vertices (Bar-Natan 1995). Bar-Natan (1995) gives a detailed account of the theory.

Expressed in simple terms, Vassiliev's fundamental idea is to study the prolongation of knot invariants to singular knots--immersions having a finite number of ordinary double points. Let denote the set of equivalence classes of singular knots with double points and no other singularities. The following definition is based on a recursion which allows to extend a knot invariant from to , then to , etc., and thus finally to the whole of . For oriented unframed knots, let be a rational-valued knot invariant. Its Vassiliev prolongation is defined by the rules

1. , and

2. Vassiliev's skein relation, illustrated below.

The right-hand side of Vassiliev's skein relation refers to the two resolutions of the double point--positive and negative. A crucial observation is that each of them is well-defined (does not depend on the plane projection used to express this relation). A knot invariant is called a Vassiliev invariant of order if its prolongation vanishes on all knots with more than ordinary double points. For example, the simplest nontrivial Vassiliev invariant has the following explicit description. Let be an arbitrary knot diagram of the given knot and an arbitrary distinguished point on , different from all crossings. Then

where the summation spreads over all pairs of crossing points such that (1) during one complete turn of the diagram in the positive direction starting from point the points and are encountered in the order , and (2) the four corresponding passages through these crossing points are underpass, overpass, overpass, and underpass, respectively. The numbers , stand for the local writhe at points and , defined according to the above illustration.

The th coefficient of the Conway polynomial is a Vassiliev invariant of order at most and, in particular, the second coefficient coincides with .

Vassiliev invariants are at least as strong as the following polynomial knot invariants: Alexander, Jones, Kauffman, and HOMFLY polynomials. This means that if two knots and can be distinguished by such a polynomial, then there is a Vassiliev invariant that takes different values for and .

The set of all -valued Vassiliev invariants , where consists of invariants of order at most , forms a vector space over the rational numbers with the increasing filtration . With , its associated graded space is a Hopf algebra. Each quotient is naturally the dual vector space of the degree- component of the unframed algebra of chord diagrams, which imposes the one-term relation (framing-independence or 1T relation) and four-term relations (Kneissler 1997, p. 3).

The dimensions of the vector spaces of Vassiliev invariants of order at most for to 12 are summarized in the following table (Kneissler 1997, p. 5; OEIS A007473).

	0	1	2	3	4	5	6	7	8	9	10	11	12
	1	1	2	3	6	10	19	33	60	104	184	316	548

The Kontsevich integral defines a universal Vassiliev invariant over the real numbers. Each finite type invariant is obtained by applying a linear functional to a finite-degree truncation of this invariant.

Duzhin (1999) records the following foundational questions about Vassiliev invariants, raised in 1990.

1. Is it true that Vassiliev invariants distinguish knots? In other words, given two nonequivalent knots and , is it always possible to indicate a finite type invariant such that ?

2. Is it true that Vassiliev invariants can detect knot orientation? More specifically, is there a knot and a finite type invariant such that , where differs from by a change of parameterization that reverses the orientation?