Added:

- Marko Berghoff,
*Graph complexes from the geometric viewpoint*, arXiv.

added pointer to today’s

- Kevin Morand,
*A note on multi-oriented graph complexes and deformation quantization of Lie bialgebroids*(arXiv:2102.07593)

added pointer to

- Marko Berghoff, Dirk Kreimer,
*Graph complexes and Feynman rules*(arXiv:2008.09540)

added pointer to Ionescu 03, Ionescu 05 regarding coproducts on graph complexes.

But what I’d really like to understand is this:

The configuration space $Conf(\mathbb{R}^n)$ has a product structure (in fact $n$ of them) given by disjoint union of configurations after moving them away from each other along one of the axes (here).

The corresponding coproduct structure on the corresponding graph complex should be that which splits off disjoint subgraphs. This is close to the coproduct considered in Ionescu 05 (2.1), but different.

The bar-cobar construction of the split-off-proper-subgraphs-coproduct should be the graph complex model for the group completion of the configuration space. Has anyone considered that?

]]>added statement of and references for the lowest order knot invariant obtained from Chern-Simons Feynman amplitudes (here)

]]>True. I found one occurrence of Peace in the entry, have changed it to “minimal trivalent diagram”.

What bothers me more is that this diagram, being closed (here), must be exact, since there is not supposed to be any cohomology in its degree.

Do you see a trivializing coboundary?

]]>Re #44 and elsewhere: not actually the Peace Symbol. More like the Mercedes-Benz logo.

]]>added this useful pointer:

- Pascal Lambrechts, Victor Tourtchine,
*Homotopy graph-complex for configuration and knot spaces*, Transactions of the AMS, Volume 361, Number 1, January 2009, Pages 207–222 (arxiv:math/0611766)

Theorem 9.3 there directly identifies the rational homotopy groups of the configuration space in terms of the cohomology of indecomposable graphs

]]>They write

“the full directed graph complex dfGC”

With your interpretation, it should be “fdGC” instead. So this can’t be how they want this to be pronounced. ;-)

]]>Not sure why you’re writing ’da’.

’d’ in ’dfGC’ is ’directed’.

]]>Sure , graph complexes with external edges do exist, modelling embeddings. That’s what I currently have in the entry, at least for some cases. The question is if they related to the *da graph complex* computing the Grothdieck-Teichmuller Lie algebra, or if the latter is a third kind of graph complex altogether.

Those ’hairy’ ones (#47) have external vertices and are used for embeddings and knot invariants.

]]>So, *da full graph complex* $dfGC$ and its siblings, relating to Grothendieck-Teichmueller, have anti-commuting edges and commuting vertices, like the graph complexes $Graphs_n(\mathbb{R}^{2D})$ modelling configuration spaces in even dimensions do. However, they don’t have any distinction between external and internal vertices. Hence if (their linear dualization) were anything like these graph models for configuration spaces, it would have to be $Graphs_0(\mathbb{R}^{2D})$ (zero external vertices) which however is trival. Also, it doesn’t look like the dualization of the differential in *da full graph complex* is the one appropriate for modeling configuration spaces and knot spaces.

So this must be an entirely different, third, class of graph complexes altogether.

]]>Not to mention hairy graphs.

]]>A bit of a mess, to my mind.

Yes! This paper has a load of variants $GC$, $fGC$, $dfGC$, each of which may have annotations to show that it is ’uncompleted’, ’1-vertex irreducible’, ’connected’.

You’d think someone in the field would look to standardize language here. Maybe a MathOverflow question is called for.

]]>added statement, proof and illustration of a simple Lemma about graphs in non-positive degree (here)

]]>I have fixed the example of the *Peace symbol* graph, now here

(It was not wrong before, but it was just zero in the graph complex, due to no internal vertices. )

Surely someone has written down these low dimension cocycles,

One would think so, wouldn’t one.

There are two concrete examples spelled out in Figures 2 and 3 in Cattaneo, Cotta-Ramusino, Longoni02. I have reproduced the computation here.

Doing so made me realize that the definition of the graph complex on their p. 18 is a little problematic. Where they say “We assume that the labelling of the external vertices is cyclic” I think what they must mean is that graphs are to be *identified* whose external labels differ by a cyclic permutation.

In any case, this is for another version of the graph complex, the one that in the entry is now called $KnotGraphs(\mathbb{R}^3)$ (here).

Here, I am instead after examples of cocycle in what in the entry is denoted $Graphs_n(\mathbb{R}^3)$, which is the complex modelling configuration spaces.

$\,$

Many examples are spelled out in Bar-Natan and McKay.

But what are these examples of? The complex they work in looks like the one called $Graphs_0(\mathbb{R}^2)$ in the entry. But this is supposed to be quasi-isomorphic to $\Omega^\bullet( Conf_0(\mathbb{R}^2) ) \simeq \Omega^\bullet( \ast ) \simeq \mathbb{R}$, so that there shouldn’t be any non-trivial examples in this case.

Indeed, as far as I see, Bar-Natan and McKay consider further filtering/bidegrees on this complex, and their examples are cocycles in particular filtering stage only. But then, what does this all mean. (?)

$\,$

I see Willwacher say …

I am still working on figuring out which exact version of the graph complex is meant in these articles. I think in these statements what is mostly meant is $KnotGraphs(\mathbb{R}^3)$, though I feel uncertain about this. In other articles it must be $Graphs(\Sigma)$ instead, but the precise details of the definition considered are not easy to extract (for me, at least).

$\,$

A bit of a mess, to my mind. Maybe some expert sees me struggling here and lends a hand. I’d be grateful!

]]>Surely someone has written down these low dimension cocycles, though I guess there something to be gained in working them out oneself.

Oh, or is it that things haven’t been worked out. I see Willwacher say about one form of Kontsevich graph complex

]]>The cohomology in positive degrees is still unknown.

Ah, there is a mistake. Both of these cocycles actually vanish. Will fix but not tonight.

]]>Similarly, here is the cocycle which I guess generates $H^0(Graphs_1(\mathbb{R}^3)) \simeq \mathbb{R}$.

Haven’t really proven yet that this is not exact. Should be easy, but it’s late now. If you see a coboundary, let me know.

]]>added mentioning that graphs without external vertices are of degree 0 precisely if they are trivalent (here) and added mentioning that the smallest of these, *Peace Symbol* graph, is a cocycle (here)

Spelled out the example (here) of the first cocycle in $KnotGraphs(\mathbb{R}^3)$ .

This computation convinces me that in the graph complex for knot space cohomology it must be understood that we are to identify graphs that differ by a cyclic ordering of the labels of their external vertices.

(This sounds very obvious, but I don’t see this stated clearly in the literature.)

]]>I have reworked the Idea-section (here) to bring out the difference between the two similar but different types of graph complexes (one for configuration spaces of points, the other for spaces of knots).

Then I started to write out the precise definition of the latter (here) but didn’t get very far before running out of time now.

]]>The difference between the graph complex for configuration spaces of points and that for knot spaces is most minor and easy to miss:

In the latter case, but not in the former, the arcs between consecutive external vertices count as contractible edges for the differential. And external vertices have degree -1 instead of 0. Everything else is the same!

It is conceivable that there is a systematic derivation of graph complex models from Weiss calculus. Volic in his articles talks about relating the two. But if there is such an explanation, I haven’t seen anyone provide any hints for it yet.

]]>