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nLab > Latest Changes: Vassiliev invariant

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    • CommentRowNumber1.
    • CommentAuthorzskoda
    • CommentTimeSep 28th 2010

    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.

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeSep 23rd 2014

    New entry Viktor Vassiliev, also redirecting V. A. Vassiliev, small updates to Picard-Lefschetz theory and Vassiliev invariant, including the redirect Vassiliev knot invariant.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeOct 2nd 2019
    • (edited Oct 2nd 2019)

    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:

    diff, v12, current

    • CommentRowNumber4.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 2nd 2019

    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.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeOct 2nd 2019

    Yeah, that’s the dots we are connecting here. But don’t tell anyone, this is top secret:

    Observables on 4-Cohomotopy space ofΣ ↓≃ cohomology of Conf(Σ,S 1) ↓≃ graph complex ofΣ ↓≃ higher Vassiliev invariants of knots inΣ ↓≃ observables of Chern-Simons theory onΣ ↓≃ observables of M5 wrapped onΣ \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 }
    • CommentRowNumber6.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 2nd 2019

    Luckily no one from outside reads here!

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeOct 2nd 2019

    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)

    diff, v14, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeOct 10th 2019
    • (edited Oct 10th 2019)

    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??

    diff, v17, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeNov 14th 2019

    Just to highlight something that only fully occurred to me now:

    That theorem by Cohen-Gitler here, saying

    χH •(ΩConf n(ℝ 3),ℂ)=∑k∈ℕdim ℂ(A k n)t k \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 ℝ 3\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 ℝ 3\mathbb{R}^3 there. But it seems it had ℝ 3\mathbb{R}^3 there all along! :-)

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeNov 22nd 2019

    added pointer to

    diff, v27, current

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeNov 22nd 2019

    added full publication data for

    • {#Kohno94} 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)

    diff, v27, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeNov 24th 2019

    added pointer to this book:

    diff, v29, current

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeDec 28th 2019

    added this reference:

    • 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 (arXiv:math/0408182)

    diff, v40, current

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeApr 30th 2020

    added pointer to today’s

    • J. de-la-Cruz-Moreno, H. García-Compeán, E. López, Vassiliev Invariants for Flows Via Chern-Simons Perturbation Theory (arXiv:2004.13893)

    diff, v45, current

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