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I have reorganized somewhat entry Maxim Kontsevich (with some new links and few bits of additional info) and created a related stub Vassiliev invariant, just to record a link to an impressive online bibliography maintained by Dror Bar-Natan and Sergei Duzhin, hosted at Duzhin's webpage at Russian Academy of Sciences.
New entry Viktor Vassiliev, also redirecting V. A. Vassiliev, small updates to Picard-Lefschetz theory and Vassiliev invariant, including the redirect Vassiliev knot invariant.
have added pointer to the following references, which express (higher order) Vassiliev invariants in terms of Feynman integrals over products of Chern-Simons propagatores assigned via the graph complex:
Maxim Kontsevich, Vassiliev’s knot invariants, Advances in Soviet Mathematics, Volume 16, Part 2, 1993 (pdf)
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)
Ismar Volić, Section 4 of: Configuration space integrals and the topology of knot and link spaces, Morfismos, Vol 17, no 2, 2013 (arxiv:1310.7224)
Should we expect that the ways in which physics illuminates knot theory to be related, e.g., Witten on the Jones polynomial and Khovanov homology, Gang-Kim-Lee on the volume conjecture, Kontevich on Vassiliev invariants, etc.? Perhaps a similar unification to your idea of twisted cohomotopy “as a grand unified theory of classical results in differential topology.”
There’s some phenomenon here that has long interested me, as I see from our conversation back here on supersymmetry and Morse theory, something about the role of physics to realize whatever it is that unifies tracts of mathematics.
Yeah, that’s the dots we are connecting here. But don’t tell anyone, this is top secret:
$\array{ \text{Observables on 4-Cohomotopy space of}\; \Sigma \\ \big\downarrow \simeq \\ \text{cohomology of }\; Conf(\Sigma,S^1) \\ \big\downarrow \simeq \\ \text{graph complex of} \; \Sigma \\ \big\downarrow \simeq \\ \text{higher Vassiliev invariants of knots in} \; \Sigma \\ \big\downarrow \simeq \\ \text{observables of Chern-Simons theory on} \; \Sigma \\ \big\downarrow \simeq \\ \text{observables of M5 wrapped on}\; \Sigma }$Luckily no one from outside reads here!
on relation to polynomial knot invariants, I have added these pointers:
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)
Myeong-Ju Jeong, Chan-Young Park, Polynomial invariants and Vassiliev invariants, Geom. Topol. Monogr. 4 (2002) 89-101 (arxiv:math/0211045)
I have added (here) the statement of Prop. 9.1 in
relating the Euler characteristic of the homology of loop spaces of configuration spaces of points to Vassiliev invariants.
But I don’t actually understand this statement yet:
They use the Euler series of the homology. Which means there is an enhancement of the grading of homology to a bigrading implicit. What’s the bigrading??
Just to highlight something that only fully occurred to me now:
That theorem by Cohen-Gitler here, saying
$\chi H_\bullet \Big( \Omega Conf_n\big( \mathbb{R}^{3} \big), \mathbb{C} \Big) \;=\; \underset{k \in \mathbb{N}}{\sum} dim_{\mathbb{C}}\big( A^n_k \big) t^k$and hence saying that the Euler-Poincare series of the homology of the loop space of the ordered configuration space is the the EP-series of the Vassiliev braid invariant
–
this theorem really has configurations in $\mathbb{R}^3$ on the left. Unless there is a typo somewhere.
I was chasing for a week for how to get a variant of the theorem that has $\mathbb{R}^3$ there. But it seems it had $\mathbb{R}^3$ there all along! :-)
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