added the example of the trefoil knot (here)

]]>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…)

]]>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 :-)

]]>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.

]]>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-diagonalentries of the linking matrix. (emphasis added)

I guess you mean that the sentence at the nLab page needs updating, then?

]]>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.

]]>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?

]]>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.

]]>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.

]]>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

]]>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. (?)

]]>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…

]]>Does this understanding of $\Omega^2 S^2$ in terms of framed links relate to Witten’s interpretation of framed links as Wilson loops?

]]>Nice! I knew there was something relevant wrt framings, but couldn’t put my finger on it.

]]>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 $+2$.

]]>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.

]]>It feels like a ribbon category minus the twist.

Or something like a framed cobordism with corners?

Very evocative, that’s for sure.

]]>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.

]]>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.

]]>I’m being reminded of John Baez on the walking ambidextrous adjunction, towards the end of twf174.

]]>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.

]]>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 $\pi_1$?

]]>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.

]]>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.)

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