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I am trying to compile references with results on configuration-space models for the iterated loop spaces $\Omega^n S^n$, $n \geq 2$.
From Segal 1973 thm 1 this is the topological group completion of the configuration space of points in $\mathbb{R}^n$ under “compress the configurations and form their disjoint union”.
The evident guess that this, in turn, should just be the configuration space of points labeled by a sign (charge) $\in \{\pm 1\}$, where oppositely charged points in the configuration are allowed to undergo pair creation/annihilation, was famously shown to be wrong (“close” but wrong) by McDuff 1975 p. 6.
A fix to this issue was claimed by Caruso & Waner 1981, who say that it works when signed points are replaced by oriented little cubes, suitably construed.
However, their definition is at least not easy to read, which made Okuyama 2004, 05 claim a “simpler” model that uses configurations of just line segments with signed endpoints subject to a clever annihilation relation.
Here is a question I have:
Okuyama’s Def. 3.3 (p. 7) specialized to $n=2$ looks to me like movements (paths) in this configuration space of line segments (intervals) in the plane can never make the intervals move around each other for the path to form a braid — contrary to what one should expect (?) in this dimension. (Namely the definition is such that the intervals stretch along the $x$-direction, say, while never being allowed to coincide in their $y$-coordinate — unless I am misreading something?)
Now this model is picked up and generalized to the equivariant setting by Okuyama & Shimakawa 2007 — however they no longer claim that it works for finite $n$ (cf. above their Th. 1.1, p. 2).
This makes me wonder if the claim in Okuyama 2004, 05 is wrong for finite $n$, or else what I am missing in my question above.
According to definition 2.2, doesn’t it look like he’s taking configuration space in the sense that one can have coinciding points?
Oh, you are right. He allows points with “summable” labels to coincide, and then later intervals inside the line are declared summable if they are disjoint.
This resolves my puzzlement. Thanks!
Am starting to make some notes – for the time being in the Sandbox.
Added an expanded graphics which makes more manifest the remarkable fact that Okuyama’s model implements just the familiar resolution of Feynman diagrams of point particles by worldsheets of strings.
(For the moment still in the Sandbox, but will give this an entry group-completed configuration spaces of points – section shortly.)
Can you say what happens in two interval endpoint collisions of the form $[0,1-\varepsilon]$ and $[1+\varespilon,2]$, and $[0,1-\varepsilon)$ and $(1+\varespilon,2]$ ? (as $\varepsilon \to 0$)? is the included/not included dichotomy meant to reflect the +ve, -ve ’charge’? Or is it a right endpoint/left endpoint thing?
The endpoints retain the same exclusion/annihilation rules as in McDuff’s model, whence I suggestively gave them the same graphical representation:
Pairs of points of equal charge may not coincide (think: they repel) while those of opposite charge may coincide and then annihilate.
Since “positive charge” in Okuyama’s description means “endpoint included in the interval”, while “negative charge” means “not included”, this implies that endpoints of closed intervals may never coincide among each other, and same for endpoints of open intervals among each other.
The only pairs of points that may coincide are the endpoints of an open half with that of a closed half of an interval:
If they belong to the same interval then them coinciding is identified with that interval (dis)appearing (into) out-of nothing, while if they belong to different intervals then them coinciding is identified with the two intervals merging into (splitting out of) the interval that is their union.
This is all stated on the first-and-a-half page of section 3 in Okuyama’s arXiv:math/0511645. It may be fun to read through his definitions there with my graphical translation in mind, which helps, I think, to see what’s going on and how beautiful this model is.
(In doing so, notice that Okuyama allows for intervals to be further labeled in a pointed topological space $X$. For the purpose of my note in the Sandbox, this space is tacitly specialized to $X \equiv S^0 = \{ 0, x\}$, so that all non-vanishing intervals carry the same label $x$, which may hence be ignored.)
Coming back to my puzzlements with understanding this model, the next one I have is this:
Writing $Conf^I(\mathbb{R}^2)$ for Okuyama’s configuration space of “charged intervals” in (specifically now) $\mathbb{R}^2$, what is a represemtative of the generator $\pm 1$ of its fundamental group, according to the following chain of equivalences (where the first is Okuyama’s and the second is Segal’s as referencd in the Sandbox, while the last one is the Hopf fibration):
$\pi_1 \big( Conf^I(\mathbb{R}^2) \big) \;\;\simeq\;\; \pi_1 \big( \mathbb{G}Conf(\mathbb{R}^2) \big) \;\;\simeq\;\; \pi_1 \big( \Omega^2 S^2 \big) \;\;\simeq\;\; \pi_0 \big( \Omega(\Omega^2 S^2) \big) \;\;\simeq\;\; \pi_0 \big( \Omega^3 S^2 \big) \;\;\simeq\;\; \pi^3(S^2) \;\;\simeq\;\; \mathbb{Z}$?
The reason I find this puzzling is that it seems easy to come up with non-trivial elements of $\pi_1 \big( Conf^I(\mathbb{R}^2) \big)$ (any link of “stringified” circles, no?) but unclear why they should all be multiples of a single generating element.
To warm up on this question, I am drawing (here) some “charged string loop diagrams” representing elements in $\pi_1$ of the Okuyama configiuration space.
It seems clear that the plain loop – with a string running on the right and an anti-string on the left – is zero in $\pi_1$ (first graphics).
How about the charged stringy Hopf link, though (second graphics). Is that a non-trivial element of $\pi_1$?
Doesn’t your diagram require three dimensions to make sense? What does the overcross/undercross mean in terms of intervals passing each other? Indeed, how could intervals pass each other in two dimensions? I feel like I’m missing something.
I’m being reminded of John Baez on the walking ambidextrous adjunction, towards the end of twf174.
Elements of $\pi_1$ of the configuration space are paths of configurations, specifically from the empty configuration to itself, whence the extra dimension.
This is analogous to the familiar situation of how a path in $Conf(\mathbb{R}^2)$ is a braid in 3d, — only that now for paths in $Conf^I(\mathbb{R}^2)$ the worldlines of the braids are thickened to string worldsheets carrying charges, and that oppositely charged strings may undergo pair creation/annihilation.
Have drawn another charged string loop diagram (here) which more readily looks like it ought to give a non-trivial element in $\pi_1$.
Will need to see how to actually prove such statements. Currently I am just staring at these diagrams and trying to envision their possible continuous deformations under the given rules.
It feels like a ribbon category minus the twist.
Or something like a framed cobordism with corners?
Very evocative, that’s for sure.
Based on the last example here I am guessing now that the element in $\mathbb{Z}$ represented by an Okuyama charged string loop is the framing number of the loop regarded as a framed link.
Have drawn a deformation of the charged stringy Hopf link (here) which suggests that it is equivalent to the framed unknot with framing number equal to $+2$.
Nice! I knew there was something relevant wrt framings, but couldn’t put my finger on it.
Does this understanding of $\Omega^2 S^2$ in terms of framed links relate to Witten’s interpretation of framed links as Wilson loops?
It seems so:
From the examples that I have illustrated so far, it seems that the class of a framed link in $\pi_1\big( Conf^I(\mathbb{R}^2)\big)$ is essentially its linking/framing number. But that linking number, in turn, may be understood essentially as the quantum observable of the corresponding Wilson loop in $U(1)$-Chern-Simons theory (e.g. p. 8 in CMNP19).
In this sense one could say that that loops in the stringy configuration space “compute” abelian Chern-Simons observables on Wilson loops.
A caveat being a factor of 2 which I still need to understand better:
The examples in the Sandbox seem to say that the $\pi_1(ConfigSpace)$-invariant of the Hopf link is 2, but of the unknot with framing number 1 is 1. Now the latter is the correct self-linking number, but the former is twice the linking number of the Hopf link.
I was guessing this is to be understood as the effect of having framings on links even when not computing self-linking. But that perspective would seem to demand the Hopf link to represent the value $4 = 2\cdot 2$ (for both framed unknots that it links each contributing with both of their boundaries) and hence still be off by a factor of 2, now in the other direction.
I should probably look at further examples in order to resolve this…
have added (here) also the example of the Hopf link with the opposite relative orientation of the links than before, showing that where the previous Hopf link had twice this unit class, this opposite one represents $-2$ times that unit class.
All these manipulations just use the one move (here) that replaces a pair of parallel but oppositely oriented strands with a cap followed by a cup. That move probably has a standard name in knot/link theory. (?)
Not quite like the skein relations, but close.
Hmm, the path that relates the two things should look like a saddle. The parallel strands you start with could be a ’rotated’ cap and cup (not literally, but they could be curved to resemble them). Feels like movie-moves stuff. See eg page 12 here: https://bpb-us-w2.wpmucdn.com/u.osu.edu/dist/5/89978/files/2021/03/CKVK-Seminar-March-1-2021.pdf.
Maybe you want ’Morse modifications’? https://www.researchgate.net/figure/Morse-Modifications-respectively-a-Minimal-Point-a-Saddle-Point-and-a-Maximal_fig1_1908314
Right, that’s the move, Martins’ “oriented saddle point transition”.
Maybe it hasn’t found much attention yet as an operation on framed link diagrams.
Regarding the factor 2 in #19:
It’s not in contradiction to anything and now I see that it’s just natural:
Namely with the linking matrix of a framed link having as off-diagonal entries the linking numbers of pairs of connected components, and as diagonal entries the framing numbers of the connected components, then the invariant that I am seeing is just the sum of all entries of the linking matrix. This is arguably the proper number to be addressed as twice the “total linking number”.
With that terminology, I think now that the theorem I am getting at is this:
Theorem. The map which regards framed oriented link diagrams as based loops in Okuyama’s configuration space $Conf^I(\mathbb{R}^2)$ descends to a map from isotopy classes to the fundamental group $\simeq \mathbb{Z}$ and as such is twice the “total linking number”, namely the sum over all entries of the linking matrix.
Moreover, the way this number comes out is by applying the “oriented saddle point transition” until a given framed link diagram is reduced to a disjoint union of framed unknots, and then taking their total framing number.
Very nice.
That said, the entry you like says
Hence for a framed link the total linking number is equivalently half the sum of all non-diagonal entries of the linking matrix.
And this is what you want, right? Not half the sum of all the entries?
No, I mean that it is natural to sum over all entries instead, and that that’s the quantity which actually appears here in the configuration space business.
As a result, the framing numbers (on the diagonal) appear with half the weight as the separate linking numbers (which appear in pairs off the diagonal) and that explains the factor of 2 I was wondering about in #19.
Well, one of these is wrong then.
From the theorem statement in #23:
twice the “total linking number”, namely the sum over all entries of the linking matrix.
From linking matrix:
the total linking number is equivalently half the sum of all non-diagonal entries of the linking matrix. (emphasis added)
I guess you mean that the sentence at the nLab page needs updating, then?
The nLab entry refers to what some author defined there a paragraph before. I am advocating here to instead regard half of the unrestricted sum of entries as the total linking number, hence including the “self-linking”. But neither of them is “wrong”, it’s just a different choice of terminology.
Yes, that was my other thought. But if you were looking for established terminology, then coming up with a new meaning for an existing phrase, even if perhaps justified, is going to need some clear flagging.
Here’s an old example of the terminology where the sum is over the non-diagonal entries: https://pi.math.cornell.edu/~mec/2008-2009/HoHonLeung/page5_knots.htm See also definition 1.1 in https://discover.wooster.edu/jbowen/files/2013/10/Total-linking-numbers-of-torus-links-and-Klein-links.pdf giving the raw definition.
So it’s not like the author of the nLab page made it up :-)
Have added (here) the graphical proof that the vacuum and saddle move imply the 1st Reidemeister move for framed links – implying that we have a well-defined map from isotopy classes of framed oriented links to the fundamental group of Okuyama’s configuration space.
(It’s becoming high time that I move out of the Sanbox into an actual entry…)
added the example of the trefoil knot (here)
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