nLab weight systems are the associated graded objects of Vassiliev invariants





(weight systems are associated graded of Vassiliev invariants)

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

𝒱 n/𝒱 n1AAAA(𝒜 n u) * \mathcal{V}_n/\mathcal{V}_{n-1} \underoverset{\simeq}{\phantom{AAAA}}{\longrightarrow} \big( \mathcal{A}_n^u \big)^\ast


  1. the quotient vector space of order-nn Vassiliev invariants of knots by those of order n1n-1

  2. to the space of unframed weight systems of order nn.

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.

Facts about chord diagrams and their weight systems:

chord diagramsweight systems
linear chord diagrams,
round chord diagrams
Jacobi diagrams,
Sullivan chord diagrams
Lie algebra weight systems,
stringy weight system,
Rozansky-Witten weight systems

chord diagram,
Jacobi diagram
horizontal chord diagram
1T&4T relation2T&4T relation/
infinitesimal braid relations
weight systemhorizontal weight system
Vassiliev knot invariantVassiliev braid invariant
weight systems are associated graded of Vassiliev invariantshorizontal weight systems are cohomology of loop space of configuration space



Last revised on April 25, 2021 at 07:15:26. See the history of this page for a list of all contributions to it.