# nLab Vassiliev invariant

Vassiliev knot invariants

### Context

#### Knot theory

knot theory

Examples/classes:

Types

knot invariants

Related concepts:

category: knot theory

# Vassiliev knot invariants

## Idea

The space of knots in the Euclidean space $\mathbb{R}^3$ (or in the 3-sphere $S^3$) 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.

###### Remark

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

### Relation to Chord diagrams and weight systems

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.

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 singular 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 “1T relation” (or “framing independence”) and the “4T relation”: see Theorem 1 of Bar-Natan 95 (or Theorem 6.2.13 of Lando & Zvonkin):

###### Proposition

(weight systems are associated graded of Vassiliev invariants)

For ground field $k = \mathbb{R}, \mathbb{C}$ the real numbers or complex numbers, there is for each natural number $n \in \mathbb{N}$ a canonical linear isomorphism

$\mathcal{V}_n/\mathcal{V}_{n-1} \underoverset{\simeq}{\phantom{AAAA}}{\longrightarrow} \big( \mathcal{A}_n^u \big)^\ast$

from

1. the quotient vector space of order-$n$ Vassiliev invariants of knots by those of order $n-1$

2. to the space of unframed weight systems of order $n$.

In other words, in characteristic zero, the graded vector space of unframed weight systems is the associated graded vector space of the filtered vector space of Vassiliev invariants.

(Bar-Natan 95, Theorem 1, following Kontsevich 93)

### Relation to homology of loop spaces of configuration spaces of points

We discuss the relation between Vassiliev invariants and the Euler characteristic of the ordinary homology of loop spaces of configuration spaces of points:

For $n, q \in \mathbb{N}$ and $q \geq 1$, write

1. $\underset{{}^{\{1,\cdots, n\}}}{Conf}\big( \mathbb{R}^{q+2} \big)$ for the configuration space of n ordered points in Euclidean space $\mathbb{R}^{q+2}$;

2. $\Omega \underset{{}^{\{1,\cdots, n\}}}{Conf}\big( \mathbb{R}^{q+2} \big)$ for the corresponding based loop space (for any choice of base point);

3. $H_\bullet\Big(\Omega \underset{{}^{\{1,\cdots, n\}}}{Conf}\big( \mathbb{R}^{q+2} \big), \mathbb{C} \Big)$ for the ordinary homology of this loop space, with coefficients in the complex numbers;

4. $\chi H_\bullet\Big(\Omega \underset{{}^{\{1,\cdots, n\}}}{Conf}\big( \mathbb{R}^{q+2} \big), \mathbb{C} \Big)$ for the Euler characteristic-series of the homology

Write also

1. $V^n_k$ for the complex vector space of Vassiliev invariants of order $k$ for pure braids with $n$ strands;;

2. $A^n_k$ for the complex vector space spanned by the horizontal chord diagrams with $n$ vertical strands modulo the “horizontal 4T relation”

such that there is an linear isomorphism

$V^n_k/V^n_{k-1} \simeq (A^n_k)^\ast$

between the quotient vector space of Vassiliev invariants and the dual vector space of chord diagrams.

Then:

The Euler characteristic-series (…) of the homology of the loop spaces of configuration spaces

$\chi H_\bullet\Big(\Omega \underset{{}^{\{1,\cdots, n\}}}{Conf}\big( \mathbb{R}^{q+2} \big), \mathbb{C} \Big) \;=\; \Big[ \big( 1 - t^q \big) \cdot \big( 1 - 2 t^q \big) \cdots \big( 1 - (n-1) t^q \big) \Big]^{-1}$

and is related to the complex dimensions of spaces of Vassiliev invariants according to

(1)$\chi H_\bullet \Big( \Omega \underset{{}^{\{1,\cdots, n\}}}{Conf}\big( \mathbb{R}^{3} \big), \mathbb{C} \Big) \;=\; \underset{k \in \mathbb{N}}{\sum} dim_{\mathbb{C}}\big( A^n_k \big) t^k$

(Cohen-Gitler 01, Prop. 9.1, based on Cohen 76 and Kohno 94)

Alternatively, we have the combination of the following two facts, via weight systems:

###### Proposition

(weight systems are cohomology of loop space of configuration space)

For ground field $k = \mathbb{R}$ the real numbers, there is a canonical injection of the real vector space $\mathcal{W}$ of framed weight systems (?) into the real cohomology of the based loop spaces of the ordered configuration spaces of points in 3-dimensional Euclidean space:

$\mathcal{W} \;\overset{\;\;\;\;}{\hookrightarrow}\; H^\bullet \Big( \Omega \underset{{}^{\{1,\cdots,n\}}}{Conf} \big( \mathbb{R}^3 \big) \Big)$

This is stated as Kohno 02, Theorem 4.2.

###### Proposition

(weight systems are associated graded of Vassiliev invariants)

For ground field $k = \mathbb{R}, \mathbb{C}$ the real numbers or complex numbers, there is for each natural number $n \in \mathbb{N}$ a canonical linear isomorphism

$\mathcal{V}_n/\mathcal{V}_{n-1} \underoverset{\simeq}{\phantom{AAAA}}{\longrightarrow} \big( \mathcal{A}_n^u \big)^\ast$

from

1. the quotient vector space of order-$n$ Vassiliev invariants of knots by those of order $n-1$

2. to the space of unframed weight systems of order $n$.

In other words, in characteristic zero, the graded vector space of unframed weight systems is the associated graded vector space of the filtered vector space of Vassiliev invariants.

(Bar-Natan 95, Theorem 1, following Kontsevich 93)

## Examples

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

(…)

### General

The original articles are

Lecture notes:

Textbook accounts:

Further review:

• Simon Willerton, On the Vassiliev invariants for knots and for pure braids, 1997 (hdl:1842/11581, ethos.663801, pdf)

• Dror Bar-Natan, Finite Type Invariants, in: J.-P. Francoise, G.L. Naber and Tsou S.T. (eds.) Encyclopedia of Mathematical Physics, Oxford: Elsevier, 2006, volume 2 page 340

• Sergei K. Lando and Alexander K. Zvonkin, Chapter 6 of: Graphs on Surfaces and Their Applications, Springer, 2004.

More literature is listed at

Concrete computations:

• Jan Kneissler, On spaces of connected graphs I: Properties of Ladders, Proc. Internat. Conf. “Knots in Hellas ‘98”, Series on Knots and Everything, vol. 24 (2000), 252-273 (arXiv:math/0301018)

• Jan Kneissler, On spaces of connected graphs II: Relations in the algebra Lambda, Jour. of Knot Theory and its Ramif. vol. 10, no. 5 (2001), 667-674 (arXiv:math/0301019)

• Jan Kneissler, On spaces of connected graphs III: The Ladder Filtration, Jour. of Knot Theory and its Ramif. vol. 10, no. 5 (2001), 675-686 (arXiv:math/0301020)

• Pierre Vogel, Algebraic structures on modules of diagrams, Journal of Pure and Applied Algebra Volume 215, Issue 6, June 2011, Pages 1292-1339 (pdf)

### Relation to Jones polynomial:

Relation to the Jones polynomial:

• Joan S. Birman; Xiao-Song Lin, Knot polynomials and Vassiliev’s invariants, Inventiones mathematicae (1993) Volume: 111, Issue: 2, page 225-270 (https://dml:144077)

Relation to other polynomial knot invariants:

• Myeong-Ju Jeong, Chan-Young Park, Polynomial invariants and Vassiliev invariants, Geom. Topol. Monogr. 4 (2002) 89-101 (arxiv:math/0211045)

### Relation to homology of loop spaces of configuration spaces

Relation to the Euler characteristic of the ordinary homology of loop spaces of configuration spaces of points

based on

• Toshitake Kohno, Vassiliev invariants and de Rham complex on the space of knots,

In: Yoshiaki Maeda, Hideki Omori and Alan Weinstein (eds.), Symplectic Geometry and Quantization, Contemporary Mathematics 179 (1994): 123-123 (doi:10.1090/conm/179)

• Fred Cohen, The homology of $\mathcal{C}_{n+1}$-Spaces, $n \geq 0$. In: The Homology of Iterated Loop Spaces. Lecture Notes in Mathematics, vol 533. Springer, Berlin, Heidelberg 1976 (doi:10.1007/BFb0080467)

### As Chern-Simons amplitudes

Discussion of higher order Vassiliev invariants as Chern-Simons theory-correlators, hence as configuration space-integrals of wedge products of Chern-Simons propagators assigned to edges of Feynman diagrams in the graph complex:

• Kontsevich 93, Section 5

• Daniel Altschuler, Laurent Freidel, Vassiliev knot invariants and Chern-Simons perturbation theory to all orders, Commun. Math. Phys. 187 (1997) 261-287 (arxiv:q-alg/9603010)

• Alberto Cattaneo, Paolo Cotta-Ramusino, Riccardo Longoni, Configuration spaces and Vassiliev classes in any dimension, Algebr. Geom. Topol. 2 (2002) 949-1000 (arXiv:math/9910139)

• Alberto Cattaneo, Paolo Cotta-Ramusino, Riccardo Longoni, Algebraic structures on graph cohomology, Journal of Knot Theory and Its Ramifications, Vol. 14, No. 5 (2005) 627-640 (arXiv:math/0307218)

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