# nLab weight systems are the associated graded objects of Vassiliev invariants

Contents

### Context

#### Knot theory

knot theory

Examples/classes:

Types

knot invariants

Related concepts:

category: knot theory

# Contents

## Statement

###### 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, Theorem 2.1)

The proof proceeds via construction of a universal Vassiliev invariant identified with the un-traced Wilson loop observable of perturbative Chern-Simons theory.

## References

Review:

Last revised on April 25, 2021 at 03:15:26. See the history of this page for a list of all contributions to it.