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

]]>added pointer to:

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

added pointer to:

- Saunders MacLane, §XI.4 of:
*Categories for the Working Mathematician*, Graduate Texts in Mathematics**5**Springer (second ed. 1997) [doi:10.1007/978-1-4757-4721-8]

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*)

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:

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]

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

]]>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)

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

]]>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]

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. )

added pointer to:

- Adolf Hurwitz, §II of:
*Über Riemann’sche Flächen mit gegebenen Verzweigungspunkten*, Mathematische Annalen**39**(1891) 1–60 [doi:10.1007/BF01199469]

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

]]>added pointer to

- Wilhelm Magnus,
*Über Automorphismen von Fundamentalgruppen berandeter Flächen*, Mathematische Annalen**109**(1934) 617–646 [doi:10.1007/BF01449158]

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

]]>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

]]>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

- Emil Artin,
*Theorie der Zöpfe*, Abh. Math. Semin. Univ. Hambg.**4**(1925) 47–72 [doi;10.1007/BF02950718]

(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]

also pointer to:

- Emil Artin,
*Theorie der Zöpfe*, Abh. Math. Semin. Univ. Hambg.**4**(1925) 47–72 [doi;10.1007/BF02950718]

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:

- Michael Friedman,
*Mathematical formalization and diagrammatic reasoning: the case study of the braid group between 1925 and 1950*, British Journal for the History of Mathematics**34**1 (2019) 43-59 [doi:10.1080/17498430.2018.1533298]

added pointer to:

- Joan S. Birman, Anatoly Libgober (eds.)
*Braids*, Contemporary Mathematics**78**(1988) [doi:10.1090/conm/078]

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

]]>completed publication data for:

- Ralph H. Fox, Lee Neuwirth,
*The braid groups*, Math. Scand.**10**(1962) 119-126 $[$doi:10.7146/math.scand.a-10518, pdf, MR150755$]$

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*

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.

]]>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?

]]>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 :-).

]]>