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the entry braid group said what a braid is, but forgot to say what the braid group is; I added in a sentence, right at the beginning (and fixed some other minor things).
This entry is not displaying well. The svg graphics do not display and something is going wrong with the the headings further down to page. I could not see what was causing this.
What is wrong with the headings? I tried what I thought would solve it but it did not have any effect.
The examples of SVG diagrams on knot and knot group work fine!
I tried the svg code in inkscape and it worked well. I created a pdf file and tried to add that to the braid group page but could not get the syntax right!
It depends a bit on the content of the svg code I think. Although knot does not look correct to me.
I think something like
[[filename.jpg:pic]]
should work. I don’t think it matters what the file extension is.
I don’t think there is anything wrong with the svg code per se, just that it is confusing the renderer. So it would be OK to have it as an svg graphic, just not with the code embedded.
added pointer to:
also these:
Joshua Lieber, Introduction to Braid Groups, 2011 (pdf)
Juan González-Meneses, Basic results on braid groups, Annales Mathématiques Blaise Pascal, Tome 18 (2011) no. 1, pp. 15-59 (ambp:AMBP_2011__18_1_15_0)
Alexander I. Suciu, He Wang, The pure braid groups and their relatives, Perspectives in Lie theory, 403-426, Springer INdAM series, vol. 19, Springer, Cham, 2017 (arXiv:1602.05291)
I have fixed the typesetting in the paragraph you added (here). The syntax errors you introduced have caused all text that followed to not render properly.
So I have:
fixed $C_n(X)
to $C_n(X)$
(missing closing dollar sign)
fixed the missing square bracket in [fat diagonal]]
fixed $\bfB_n(X)$
to $\mathbf{B}_n(X)$
(though the problem remains that neither matches the notation used elsewhere in the entry)
Also I have:
changed “permutation group” to “symmetric group”,
fixed [[morse theory]]
to [[Morse theory]]
Beyond that, I suggest more edits:
The paragraph you added really does two independent things:
1) It (almost) says that in the guise of the fundamental group of a configuration space of points, braid groups can be defined for general topological spaces. This could well we part of the Definition-section right after the “Geometric definition”. One should then start out making this explicit, with words like “The above definition of the braid group as the fundamental group of a configuration spaces of points in Euclidean space immediatly generalizes to other topological spaces. …”
2) It claims some special properties for the case where the topological space is a graph. This should go to “Examples” or maybe “Properties”. But currently this comment remains vague and may need to be expanded to be useful. Could you give a pointer to the reference that you are alluding to here (“An and Maciazek”)?
Finally, allow me suggest to avoid introducing a definition by the words “there is currently much interest”. First, the text added may stay around for years or forever, so that instead of “currently” one needs to say “as of 2021” or similar for it to keep making sense. Second, if there really is much interest, then it should be easy to give a couple of review references, which would be more useful for the reader. But mainly I think that “there is currently much interest” is a lame way of justifying content by appeal to the perceived authority of an unspecified group, where it would be more useful to justify it by mentioning of some of its intrinsic merits.
Where it says
Elements of the braid group $\bfB_n(X)$
aside from the bf issue, how are these braid groups to be seen in comparison to other braid groups mentioned on the page?
Are all other mentions on the page the same group? We have $Br_n$, $B_n$, $Braids_{2k}$ seemingly referring to the same groups.
The entry starts off with the braids in 3-dimensional Cartesian space. Then we get to the definition and it’s about the configuration space of points in $\mathbb{C}^n$.
I have tried to harmonize the notation for the group, making it “$Br(n)$” and “$PBr(n)$” throughout.
On the question in #13: A path of configurations of points in the plane is a braid in 3d space! I have added a sentence at the beginning of “Geometric definition” (here) to bring out the informal idea right up front. (Not claiming this couldn’t be expanded on.)
Then I went and partially rewrote (here) the addition from #10 as suggested in #11.
It is possible to give a geometric definition in $\mathbb{R}^{3}$, but once one has been able to find a good algebraic formalism, I’d typically view the geometric definition informally, i.e. as a form of pictorial notation which ultimately/in principle can be translated back to algebra. I think one typically ends up in a minefield of needless clutter, à la Joyal and Street, if one tries to make the geometric manipulations ’rigorous’. This becomes very relevant one-dimension up: the geometry of 2-braids and things like 2-Temperley-Lieb algebras is very intricate, and I definitely think it is better to fix some algebraic (i.e. category/higher category theoretic) definition and use a pictorial notation to informally reason about it than to try to do a Joyal-Street one dimension up.
I’d certainly favour this geometric/algebraic point of view as primary over the configuration space one, but that may well just be a matter of mathematical taste :-).
Re #14, thanks, that’s clearer.
Now, the examples are rendering badly. I guess a simple list would be adequate, or other pages calling to these numbered examples?
I have fixed the formatting of the examples (here), also expanded slightly and added some hyperlinks to keywords.
I think it’s good practice, especially on a collaborative wiki, to include every thought in its numbered environment. The more this is done, the more pages are modular and robustly accessible under all kinds of future edits.
I finally realized that there had been a paragraph on surface braid groups already at the very end. I have partially merged that into the edit of #14 and partially made it a further Example: Hurwitz braid group
completed publication data for:
added pointer to:
added pointer to the original articles:
Emil Artin, Theory of Braids, Annals of Mathematics, Second Series, 48 1 (1947) 101-126 [doi:10.2307/1969218]
Frederic Bohnenblust, The Algebraical Braid Group, Annals of Mathematics Second Series 48 1 (1947) 127-136 [doi:10.2307/1969219]
Wei-Liang Chow, On the Algebraical Braid Group, Annals of Mathematics Second Series, 49 3 (1948) 654-658 [doi:10.2307/1969050]
and to this historical account:
also pointer to:
I have touched the notation in the section “Group-theoretic definition” (here) in an attempt to make it more reader-frinedly (still room left in this direction, though).
(By the way, I am inlinded to remove the sub-sectioning into “Geometric definition” and “Group-theoretic defintion”: It’s hard and probably pointless to maintain such a distinction, and in fact it has been violated since rev 15, when the MCG-characterization was inserted within the latter instead of the former.)
Then I have started to add references for the statements relating to automorphisms of free groups: Besides pointer to section 6 in the original
(which, amazingly, still trumps most accounts in terms of illustrating graphics – I’ll copy some of them into the entry) I have so far added pointer to
Lluís Bacardit, Warren Dicks, Actions of the braid group, and new algebraic proofs of results of Dehornoy and Larue, Groups Complexity Cryptology 1 (2009) 77-129 [arXiv:0705.0587, doi;10.1515/GCC.2009.77]
Valerij G. Bardakov, Extending representations of braid groups to the automorphism groups of free groups, Journal of Knot Theory and Its Ramifications 14 08 (2005) 1087-1098 [arXiv:math/0408330, doi:10.1142/S0218216505004251]
added pointer to
(first discussion of the braid group as a mapping class group)
added pointer to:
(where the braid group already appears as the fundamental group of a configuration space of points – albeit neither under these names).
have produced tikz
-graphics for the Artin presentation: here
(I keep being surprised, though, how the nLab’s tikz
-rendering differs from that on my local machine: a yscale=.5
-flag to a tikzpicture
on my local installation scales the coordinate positions, but not for instance the font — but now in the nLab the font comes up stunted, too. )
adding references with presentations of the pure braid group:
Dan Margalit, Jon McCammond, Geometric presentations for the pure braid group, Journal of Knot Theory and Its Ramifications 18 01 (2009) 1-20 [arXiv:math/0603204, doi:10.1142/S0218216509006859]
Eon-Kyung Lee, A positive presentation of the pure braid group, Journal of the Chungcheong Mathematical Society 23 3 (2010) 555-561 [JAKO201007648745187, pdf]
added pointer to:
Jennifer C. H. Wilson, The geometry and topology of braid groups, lecture at 2018 Summer School on Geometry and Topology, Chicago (2018) [pdf]
Jennifer C. H. Wilson, Representation stability and the braid groups, talk at ICERM – Braids (Feb 2022) [pdf]
have added tikz
-pictures (here) of all the relations for the presentation of the pure braid group — in terms of Artin’s generators, but using the optimized set of relations from Lee (2010)
unfortunately, only after drawing these I realized that Lee silently changes the convention how generators match to pictures in going from her Figure 1 to Figure 2. As a result, the diagrams I have drawn now correspond, strictly speaking, to the relations on the inverses of Lee’s generators.
I’ll look into harmonizing this, but not right now.
added pointer to:
Frederick R. Cohen, Braid orientations and bundles with flat connections, Inventiones mathematicae 46 (1978) 99–110 [doi:10.1007/BF01393249]
Jonathan Beardsley, On Braids and Cobordism Theories Glasgow (2022) [notes: pdf]
(also at G-structure)
added pointer to:
at the end of the section on braid groups as mapping class groups, I have added a brief remark (here) highlighting that therefore the braid group acts canonically on the fundamental group of the punctured surface (with a couple of furhter references) thus logically leading over to the following section
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