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nLab > Latest Changes: braid group

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMay 14th 2012

    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).

    • CommentRowNumber2.
    • CommentAuthorTim_Porter
    • CommentTimeOct 5th 2018

    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.

  3. Thanks for raising this, Tim. The problem is the svg code that is put directly in the entry. I would suggest to extract the code for each graphic into a file, and then upload it as an svg image to the nLab, using the usual syntax for images.

    I do not have time myself just now, but can do it when I get the chance if no-one manages it beforehand.
    • CommentRowNumber4.
    • CommentAuthorTim_Porter
    • CommentTimeOct 5th 2018
    • (edited Oct 5th 2018)

    What is wrong with the headings? I tried what I thought would solve it but it did not have any effect.

  5. The svg code confuses the renderer, I believe.
    • CommentRowNumber6.
    • CommentAuthorTim_Porter
    • CommentTimeOct 5th 2018

    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!

    • CommentRowNumber7.
    • CommentAuthorRichard Williamson
    • CommentTimeOct 5th 2018
    • (edited Oct 5th 2018)

    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.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeMay 20th 2021

    added pointer to:

    diff, v28, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeMay 20th 2021

    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)

    diff, v29, current

    • CommentRowNumber10.
    • CommentAuthorGrant_Bradley
    • CommentTimeJun 24th 2021

    Preliminaries of geometric presentations of braid groups for particles on a graph, attached reference

    diff, v33, current

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJun 24th 2021
    • (edited Jun 24th 2021)

    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.

    diff, v34, current

    • CommentRowNumber12.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 24th 2021

    Where it says

    Elements of the braid group bfB n(X)\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 nBr_n, B nB_n, Braids 2kBraids_{2k} seemingly referring to the same groups.

    • CommentRowNumber13.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 24th 2021

    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 ℂ n\mathbb{C}^n.

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeJun 24th 2021
    • (edited Jun 24th 2021)

    I have tried to harmonize the notation for the group, making it “Br(n)Br(n)” and “PBr(n)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.

    diff, v35, current

    • CommentRowNumber15.
    • CommentAuthorRichard Williamson
    • CommentTimeJun 24th 2021
    • (edited Jun 24th 2021)

    It is possible to give a geometric definition in ℝ 3\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 :-).

    • CommentRowNumber16.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 24th 2021

    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?

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeJun 24th 2021

    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.

    diff, v37, current

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeJun 24th 2021

    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

    diff, v37, current

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeJun 25th 2022

    completed publication data for:

    diff, v40, current

    • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeJun 25th 2022

    Under “Geometric definition” (here) I have added pointer to references

    diff, v40, current

    • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeAug 13th 2022

    added pointer to:

    diff, v41, current

    • CommentRowNumber22.
    • CommentAuthorUrs
    • CommentTimeSep 20th 2022
    • (edited Sep 20th 2022)

    added pointer to the original articles:

    and to this historical account:

    diff, v42, current

    • CommentRowNumber23.
    • CommentAuthorUrs
    • CommentTimeSep 20th 2022

    also pointer to:

    diff, v43, current

    • CommentRowNumber24.
    • CommentAuthorUrs
    • CommentTimeSep 20th 2022
    • (edited Sep 20th 2022)

    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

    diff, v43, current

    • CommentRowNumber25.
    • CommentAuthorUrs
    • CommentTimeSep 20th 2022

    have added more comments on how the braid group is a mapping class group (now here)

    In the process, I have re-arranged the ambient “Definition”-section: Now it is called “Definitions and Characterizations” (here) and has four subsections at equal depth

    diff, v44, current

    • CommentRowNumber26.
    • CommentAuthorUrs
    • CommentTimeSep 20th 2022

    added pointer to

    (first discussion of the braid group as a mapping class group)

    diff, v45, current

    • CommentRowNumber27.
    • CommentAuthorUrs
    • CommentTimeSep 20th 2022

    added pointer to:

    (where the braid group already appears as the fundamental group of a configuration space of points – albeit neither under these names).

    diff, v45, current

    • CommentRowNumber28.
    • CommentAuthorUrs
    • CommentTimeFeb 4th 2023

    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. )

    diff, v49, current

    • CommentRowNumber29.
    • CommentAuthorUrs
    • CommentTimeFeb 5th 2023

    adding references with presentations of the pure braid group:

    diff, v50, current

    • CommentRowNumber30.
    • CommentAuthorUrs
    • CommentTimeFeb 5th 2023

    replaced the lead-in sentence of the Idea and the first graphics with something better (here)

    diff, v51, current

    • CommentRowNumber31.
    • CommentAuthorUrs
    • CommentTimeFeb 5th 2023

    have now re-worked the entire Idea-section (here). What was good about the diagrams shown there previously is meanwhile better rendered in the following section Via generators and relations (here)

    diff, v51, current

    • CommentRowNumber32.
    • CommentAuthorUrs
    • CommentTimeFeb 5th 2023

    added discussion of finite presentation of the pure braid group: here

    diff, v51, current

    • CommentRowNumber33.
    • CommentAuthorUrs
    • CommentTimeFeb 5th 2023

    added pointer to:

    diff, v51, current

    • CommentRowNumber34.
    • CommentAuthorUrs
    • CommentTimeFeb 10th 2023
    • (edited Feb 10th 2023)

    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.

    diff, v53, current

    • CommentRowNumber35.
    • CommentAuthorUrs
    • CommentTimeFeb 26th 2023

    added pointer to:

    (also at G-structure)

    diff, v57, current

    • CommentRowNumber36.
    • CommentAuthorUrs
    • CommentTimeMay 20th 2023

    added pointer to:

    diff, v61, current

    • CommentRowNumber37.
    • CommentAuthorUrs
    • CommentTimeJul 5th 2023

    added pointer to:

    • Marie Abadie, §1 in: A journey around mapping class groups and their presentations (2022) [pdf]

    diff, v63, current

    • CommentRowNumber38.
    • CommentAuthorUrs
    • CommentTimeJul 5th 2023

    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

    diff, v63, current

    • CommentRowNumber39.
    • CommentAuthorUrs
    • CommentTimeOct 28th 2024

    made some cosmetic adjustments in the section As mapping class group of punctured disk (here)

    diff, v66, current

1 to 39 of 39

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