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nLab > Latest Changes: weight systems are cohomology of knot graph complex

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeDec 10th 2019

    starting something, to record Prop. 7.6 in

    • 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)

    still need to add details – not done yet

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeDec 11th 2019

    have spelled out the statement in mode detail (here)

    diff, v3, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeDec 11th 2019

    have streamlined the notation, in order to better bring out how neat the statement is:

    ⟨Pexp(∫ (−)A)⟩≔∑n∈ℕℏ n∑Γ∈𝒟 n cs(1|Γ|[Γ] 𝒜⊗[Γ] 𝒢) ∈H 0(𝒜 •⊗𝒢 •) ≃Hom(𝒲 •,H •(𝒢)) \begin{aligned} \left\langle \text{P}\exp \left( \int_{(-)} A \right) \right\rangle \;\coloneqq\; \underset{ n \in \mathbb{N} }{\sum} \hbar^n \underset{ \Gamma \in \mathcal{D}^{cs}_n }{\sum} \left( \frac{1}{\left\vert \Gamma\right\vert} \, [\Gamma]_{\mathcal{A}} \otimes [\Gamma]_{\mathcal{G}} \right) \; & \in \;\; H^0 \big( \mathcal{A}_\bullet \otimes \mathcal{G}^\bullet \big) \\ & \;\simeq\; Hom \big( \mathcal{W}^\bullet , H^\bullet(\mathcal{G}) \big) \end{aligned}

    diff, v4, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeDec 11th 2019

    In fact it’s better still, but maybe the full beauty is not so easily brought out with Instiki.

    For the moment I have included graphics of a LaTeX-account of the broader situation: here

    diff, v5, current

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 12th 2019

    What is the definition of the ϕ\phi as in ϕ(w)∈Graphs\phi(w) \in Graphs in the Idea section?

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeDec 12th 2019

    Ah, that was my original name for the iso in the statement. After #3 this became named ⟨Tr wPexp(∫ (−)A)⟩\left\langle Tr_{{}_{w}} \text{P}\exp \left( \int_{(-)} A \right) \right\rangle. Fixed now.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeDec 18th 2019
    • (edited Dec 18th 2019)

    added comments on how the remaining statement follows, that the comparison map is an iso

    changed page name to a permutation of the previous version that better matches naming of related statements such as weight systems are associated graded of Vassiliev invariants

    diff, v8, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeDec 26th 2019

    added this pointer on the universal Vassiliev invariant of the unknot:

    diff, v10, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeDec 26th 2019

    added this pointer on the universal Vassiliev invariant of the unknot:

    diff, v10, current

1 to 9 of 9

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