Examples/classes:
Types
Related concepts:
(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
from
the quotient vector space of order-$n$ Vassiliev invariants of knots by those of order $n-1$
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.
Facts about chord diagrams and their weight systems:
chord diagrams | weight systems |
---|---|
linear chord diagrams, round chord diagrams Jacobi diagrams, Sullivan chord diagrams | Lie algebra weight systems, stringy weight system, Rozansky-Witten weight systems |
Maxim Kontsevich, Vassiliev’s knot invariants, Advances in Soviet Mathematics, Volume 16, Part 2, 1993 (pdf)
Joan S. Birman, Xiao-Song Lin, Knot polynomials and Vassiliev’s invariants, Invent Math (1993) 111: 225 (doi:10.1007/BF01231287)
Dror Bar-Natan, On the Vassiliev knot invariants, Topology Volume 34, Issue 2, April 1995, Pages 423-472 (doi:10.1016/0040-9383(95)93237-2, pdf)
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.