Braids Nlab Article

Contents

Idea

By a braid group [Artin (1925)] one means the group of joint continuous motions [cf. Goldsmith (1981)] of a fixed number n+1n+1 of non-coincident points in the plane, from any fixed configuration back to that fixed configuration. The “worldlines” traced out by such points in space-time under such an operation look like a braid with n+1n+1 strands, whence the name.

As with actual braids, here it is understood that two such operations are identified if they differ only by continuous deformations of the “strands” without breaking or intersecting these, hence that one identifies those such systems of worldlines which are isotopic in ℝ 3\mathbb{R}^3. (This is just the kind of invariance considered for link diagrams — such as under Reidemeister moves — and in fact every link diagram may be obtained by “closing up” a braid diagram, in the evident sense.)

A quick way of saying this with precision is to observe that a braid group is thus the fundamental group π 1\pi_1 of a configuration space of points in the plane (see below for more).

Here it makes a key difference whether:

• one considers the points in a configuration as ordered (labeled by numbers 1,⋯,n+11, \cdots, n+1) in which case one speaks of the pure braid group,

• or as indistinguishable (albeit in any case with distinct positions!) in which case one speaks of the braid group proper.

Namely, after traveling along a general braid bb the order of the given points may come out permuted by a permutation perm(b)\mathrm{perm}(b), and the braid is called pure precisely if this permutation is trivial:

More explicitly, braid groups admit finite presentations by generators and relations. To these we now turn first:

Definitions and characterizations

• Via generators and relations

• As the fundamental group of a configuration space of points

• As mapping class group of punctured surfaces

• As automorphisms of a free group

Via generators and relations

Presentation of general braids

We discuss the braid group as a finitely generated group (Artin 1925, (5)-(6); Artin 1947, (18)-(19); review in, e.g.: Fox & Neuwirth 1962, §7):

Presentation of pure braids

Similarly, the pure braid group has a finite presentation.

One possible set of generators (also originally considered by Artin) are the “weave” braids where one strands lassos exactly one other strand:

In terms of these generators, the pure braid group is obained by quotienting out the following relations — an optimization of Artin’s original pure braid relations, due to Lee (2010, Thm. 1.1, Rem. 3.1):

Here we repeatedly used products of pure braid generators of the following form:

As fundamental group of a configuration space of points

Geometrically, one may understand the group of braids in ℝ 3\mathbb{R}^3 as the fundamental group of the configuration space of points in the plane ℝ 2\mathbb{R}^2 (traditionally regarded as the complex plane ℂ\mathbb{C} in this context, though the complex structure plays no role in the definition of the braid group as such).

(originally due to Hurwitz 1891, §II, then re-discovered/re-vived in Fadell & Neuwirth 1962, p. 118, Fox & Neuwirth 1962, §7, review includes Birman 1975, §1, Williams 2020, pp. 9)

In the plane

We say this in more detail:

Let C n↪ℂ nC_n \hookrightarrow \mathbb{C}^n denote the space of configurations of n ordered points in the complex plane, whose elements are those n-tuples (z 1,…,z n)(z_1, \ldots, z_n) such that z i≠z jz_i \neq z_j whenever i≠ji \neq j. In other words, C nC_n is the complement of the fat diagonal:

C n≔ℂ n∖Δ ℂ n. C_n \;\coloneqq\; \mathbb{C}^n \setminus \mathbf{\Delta}^n_{\mathbb{C}} \,.

The symmetric group S nS_n acts on C nC_n by permuting coordinates. Let:

• C n/S nC_n/S_n denote the quotient by this group action, hence the orbit space (the space of nn-element subsets of ℂ\mathbb{C} if one likes),

• [z 1,…,z n][z_1, \ldots, z_n] denote the image of (z 1,…,z n)(z_1, \ldots, z_n) under the quotient coprojection π:C n→C n/S n\pi \colon C_n \to C_n/S_n (i.e. its the equivalence class).

We understand p=(1,2,…,n)p = (1, 2, \ldots, n) as the basepoint for C nC_n, and [p]=[1,2,…n][p] = [1, 2, \ldots n] as the basepoint for the configuration space of unordered points C n/S nC_n/S_n, making it a pointed topological space.

Evidently a braid β\beta is represented by a path α:I→C n/S n\alpha: I \to C_n/S_n with α(0)=[p]=α(1)\alpha(0) = [p] = \alpha(1). Such a path may be uniquely lifted through the covering projection π:C n→C n/S n\pi: C_n \to C_n/S_n to a path α˜\tilde{\alpha} such that α˜(0)=p\tilde{\alpha}(0) = p. The end of the path α˜(1)\tilde{\alpha}(1) has the same underlying subset as pp but with coordinates permuted: α˜(1)=(σ(1),σ(2),…,σ(n))\tilde{\alpha}(1) = (\sigma(1), \sigma(2), \ldots, \sigma(n)). Thus the braid β\beta is exhibited by nn non-intersecting strands, each one connecting an ii to σ(i)\sigma(i), and we have a map β↦σ\beta \mapsto \sigma appearing as the quotient map of an exact sequence

1→PBr(n)→Br(n)→Sym(n)→11 \to PBr(n) \to Br(n) \to Sym(n) \to 1

which is part of a long exact homotopy sequence corresponding to the fibration π:C n→C n/S n\pi \colon C_n \to C_n/S_n.

In general surfaces or graphs

Since the notion of a configuration space of points makes sense for points in any topological space, not necessarily the plane ℝ 2\mathbb{R}^2, the above geometric definition has an immediate generalization:

For Σ\Sigma any surface, the fundamental group of the (ordered) configuration space of points in Σ\Sigma may be regarded as generalized (pure) braid group, called a surface braid group. (See the spherical braid group for the case that the surface in question is the 2-sphere.)

These surface braid groups are of interest in 3d topological field theory and in particular in topological quantum computation where it models non-abelian anyons.

Yet more generally, one may consider the fundamental group of the configuration space of points of any topological space XX.

For example for XX a 1-dimensional CW-complex, hence an (undirected) graph, one speaks of graph braid groups (e.g. Farley & Sabalka 2009).

The following should maybe not be here in the Definition-section, but in some Properties- or Examples-section, or maybe in a dedicated entry on graph braid groups?:

It has been shown (An & Maciazek 2006, using discrete Morse theory and combinatorial analysis of small graphs) that graph braid groups are generated by particular particle moves with the following description:

1. Star-type generators: exchanges of particle pairs on vertices of the particular graph

2. loop type generators: circular moves of a single particle around a simple cycle of the graph

Braid groups as mapping class groups of punctured surfaces

The braid groups are equivalently given by mapping class groups of punctured surfaces. [Birman 1969]

For plain braid groups

The braid group Br(n)Br(n) may be alternatively described as the mapping class group of a 2-disk D 2D^2 with nn punctures. [Birman 1969]

(review includes Birman 1975 §4, González-Meneses 2011 §1.4, Massuyeau 2021 §3.3, Abadie 2022 §1.3)

Concretely, consider

1. D 2∖{z 1,⋯,z n}D^2 \setminus \{z_1, \cdots, z_n\}

   denoting the complement of nn distinct points in the closed disk (with boundary the circle);

2. Homeo ∂(D 2∖{z 1,⋯,z n})Homeo^{\partial}\big(D^2 \setminus \{z_1, \cdots, z_n\} \big)

   denoting the mapping space of auto-homeomorphisms which restrict to the identity on the boundary circle, regarded with its canonical group structure under composition;

3. Homeo id ∂(D 2∖{z 1,⋯,z n})Homeo^{\partial}_id\big(D^2 \setminus \{z_1, \cdots, z_n\} \big)

   denoting the subgroup which is the connected component of the identity (which is readily seen to be a normal subgroup).

Then the mapping class group is the quotient group:

MCG(D 2∖{z 1,⋯,z n})≔Homeo ∂(D 2∖{z 1,⋯,z n})/Homeo id ∂(D 2∖{z 1,⋯,z n}). MCG \big( D^2 \setminus \{z_1, \cdots, z_n\} \big) \;\coloneqq\; Homeo^{\partial}\big(D^2 \setminus \{z_1, \cdots, z_n\} \big) \Big/ Homeo^{\partial}_{id}\big(D^2 \setminus \{z_1, \cdots, z_n\} \big) \,.

Now observe that

1. for the case that n=0n = 0 this group is trivial, by Alexander's trick.

2. continuous extension yields an injection

   Homeo ∂(D 2∖{z 1,⋯,z n})↪−ι−Homeo ∂(D 2). Homeo^{\partial}\big(D^2 \setminus \{z_1, \cdots, z_n\} \big) \xhookrightarrow{\phantom{-}\iota\phantom{-}} Homeo^{\partial}\big(D^2\big) \,.

Combining this implies that for every [ϕ]∈MCG(D 2∖{z 1,⋯,z n})[\phi] \,\in\, MCG\big(D^2 \setminus \{z_1, \cdots, z_n\}\big) there is an isotopy to the identity, ι[ϕ]→id\iota[\phi] \to id, under which the locations of the punctures trace out a braid (in the sense of a loop in the symmetrized configuration space of points). This construction constitutes a group homomorphism from the mapping class group to the braid group

MCG(D 2∖{z 1,⋯,z n})→∼Br(n). MCG \big( D^2 \setminus \{z_1, \cdots, z_n\} \big) \xrightarrow{\;\;\sim\;\;} Br(n) \,.

and this is an isomorphism.

For surface braid groups

More generally, let Σ\Sigma be a (connected) closed, oriented surface, possibly with boundary and let x 1,⋯,x n∈Σx_1, \cdots, x_n \in \Sigma a finite set of interior points.

Then:

(Going back to Birman 1969, cf. Massuyeau 2021 Thm. 3.13).

(In this form the statement appears in Farb & Margalit 2012, Thm. 9.1.)

In words: The mapping class group of a punctured surface is a group extension of the mapping class group of the plain surface by the surface braid group on the set of punctures.

For conditions under which Homeo 0 +,∂(Σ)=1Homeo^{+,\partial}_0(\Sigma) = 1 see here at diffeomorphism group.

As automorphisms of a free group

Since the fundamental group of D 2∖{z 1,⋯,z n}D^2 \setminus \{z_1, \cdots, z_n\} is the free group of nn generators, the MCG-presentation of the braid group (above) induces a group homomorphism of the braid group into the automorphism group of a free group, which turns out to be faithful.

This presentation is due to Artin 1925, §6, review includes González-Meneses 2011, §1.6, see also pointer in Bardakov 2005, p. 2.

More in detail, since the homotopy type of this punctured disk is, evidently, that of the the wedge sum of nn circles, it follows that its fundamental group is is the free group F nF_n on nn generators:

π 1(D 2∖{z 1,⋯,z n})≃π 1(∨ nS 1)≃F n. \pi_1 \big( D^2 \setminus \{z_1, \cdots, z_ n\} \big) \;\simeq\; \pi_1 \big( \vee_n S^1 \big) \;\simeq\; F_n \,.

Now the functoriality of π 1:Top */⟶Grp\pi_1 \,\colon\, Top^{\ast/} \longrightarrow Grp implies we have an induced homomorphism

Aut(D 2∖{z 1,⋯,z n})⟶Aut(π 1(D 2∖{z 1,⋯,z n}))≃Aut(F n). Aut \big( D^2 \setminus \{z_1, \cdots, z_n\} \big) \longrightarrow Aut \Big( \pi_1\big( D^2 \setminus \{z_1, \cdots, z_n\} \big) \Big) \,\simeq\, Aut(F_n) \,.

If such an automorphism ϕ\phi of D 2∖{z 1,⋯,z n}D^2 \setminus \{z_1, \cdots, z_n\} is isotopic to the identity, then of course π 1(ϕ)\pi_1(\phi) is trivial, which means that the above homomorphism factors through the quotient group known as the mapping class group:

MCG(D 2∖{z 1,⋯,z n})=Aut(D 2∖{z 1,⋯,z n})/Aut 0(D 2∖{z 1,⋯,z n}) MCG \big( D^2 \setminus \{z_1, \cdots, z_n\} \big) \,=\, Aut \big( D^2 \setminus \{z_1, \cdots, z_n\} \big)/Aut_0 \big( D^2 \setminus \{z_1, \cdots, z_n\} \big)

Therefore, the above gives a homomorphism of the following form, which turns out to be a monomorphism:

Br(n)=MCG(D 2∖{z 1,⋯,z n})↪−−Aut(F n) Br(n) \,=\, MCG \big( D^2 \setminus \{z_1, \cdots, z_n\} \big) \xhookrightarrow{\phantom{--}} Aut(F_n)

Explicitly, the generator yb iyb_i (1) in the Artin presentation(Def. ) is mapped to the automorphism σ i\sigma_i on the free group on nn generators t 1,…,t nt_1, \ldots, t_n which is given as follows:

σ i(t j)={t i+1 | j=i t i+1 −1⋅t i⋅t i+1 | j=i+1 t j | otherwise. \sigma_i(t_j) \,=\, \left\{ \begin{array}{lcl} t_{i+1} & \vert & j = i \\ t_{i+1}^{-1} \cdot t_i \cdot t_{i+1} & \vert & j = i + 1 \\ t_j &\vert& \text{otherwise.} \end{array} \right.

(Artin 1925, (24))

Properties

Relation to moduli space of monopoles

(Cohen-Cohen-Mann-Milgram 91)

Examples

The first few examples of the braid group Br(n)Br(n) for low values of nn:

• framed braid group

• spherical braid group

• annular braid group

• plat closure

• braid representation

• braid group statistics

  anyons, quantum Hall effect

  topological quantum computation

• braid group cryptography

• braid category

• infinitesimal braid relation

• loop braid group

• parenthesized braid operad

• braid cobordism

• vine monoid

• tangle

References

General

The braid group regarded as the fundamental group of a configuration space of points is considered (neither of them under these names, though) already in:

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

there regarded as acting on Riemann surfaces forming branched covers, by movement of the branch points.

The original articles dedicated to analysis of the braid group:

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

  (the braid group via generators & relations and via automorphisms of free groups)

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

  (the braid group as a mapping class group)

• 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]

Identifying the center of the braid group as the subgroup generated by the square of the “fundamental braid” or “half-twist”:

• F. A. Garside: The braid group and other groups, The Quarterly Journal of Mathematics 20 1 (1969) 235–254 [doi:10.1093/qmath/20.1.235, pdf, pdf]

Survey of the early history:

• Michael Friedman (historian), 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]

The understanding of the braid group as the fundamental group of a configuration space of points was re-discovered/re-vived (after Hurwitz 1891) in:

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

• Edward Fadell, Lee Neuwirth, Configuration spaces, Math. Scand. 10 (1962) 111-118 [[doi:10.7146/math.scand.a-10517, MR141126]]

The understanding of the braid group as the mapping class group of the punctured disk, etc., is due to:

• Joan S. Birman: Mapping class groups and their relationship to braid groups, Communications on Pure and Applied Mathematics 22 2 (1969) 213-238 [doi:10.1002/cpa.3160220206]

Textbook accounts:

• Joan S. Birman, Braids, links, and mapping class groups, Princeton Univ Press (1975) [ISBN:9780691081496, preview pdf]

• 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]

• Tomotada Ohtsuki ch 2 of: Quantum Invariants – A Study of Knots, 3-Manifolds, and Their Sets, World Scientific (2001) [doi:10.1142/4746]

• Seiichi Kamada: Classical Braids and Links, Part 1 of: Braid and Knot Theory in Dimension Four, Mathematical Surveys and Monographs 95, AMS (2002) [ISBN:978-0-8218-2969-1]

• Christian Kassel, Vladimir Turaev, Braid Groups, GTM 247 Springer Heidelberg 2008 (doi:10.1007/978-0-387-68548-9, webpage)

In relation to mapping class groups:

• Benson Farb, Dan Margalit: Braid groups, chapter 9 of: A primer on mapping class groups, Princeton Mathematical Series, Princeton University Press (2012) [ISBN:9780691147949, jstor:j.ctt7rkjw, pdf]

Further introduction and review:

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

• Chen Ning Yang, M. L. Ge (eds.), Braid Group, Knot Theory and Statistical Mechanics, Advanced Series in Mathematical Physics 9, World Scientific (1991) [doi:10.1142/0796]

• Dale Rolfsen: Tutorial on the braid groups, lecture notes (2007) [arXiv:1010.4051]

• Joshua Lieber, Introduction to Braid Groups, 2011 (pdf)

• Juan González-Meneses, Basic results on braid groups, Annales Mathématiques Blaise Pascal, 18 1 (2011) 15-59 [ambp:AMBP_2011__18_1_15_0, numdam:AMBP_2011__18_1_15_0]

• Alexander I. Suciu, He Wang, The pure braid groups and their relatives, Perspectives in Lie theory, INdAM series, 19 Springer (2017) 403-426 [arXiv:1602.05291]

• Dale Rolfsen: New developments in the theory of Artin’s braid groups, Topology and its Applications 127 1–2 (2003) 77-90 [doi:10.1016/S0166-8641(02)00054-8, pdf]

• John Guaschi, Daniel Juan-Pineda: A survey of surface braid groups and the lower algebraic K-theory of their group rings [arXiv:1302.6536]

• Jennifer C. H. Wilson, The geometry and topology of braid groups, lecture at 2018 Summer School on Geometry and Topology, Chicago (2018) [pdf, pdf]

• Lucas Williams, Configuration Spaces for the Working Undergraduate, Rose-Hulman Undergraduate Mathematics Journal, 21 1 (2020) Article 8 [arXiv:1911.11186, rhumj:vol21/iss1/8]

• Gwénaël Massuyeau: Lectures on Mapping Class Groups, Braid Groups and Formality (2021) [pdf, pdf]

• Jennifer C. H. Wilson, Representation stability and the braid groups, talk at ICERM – Braids (Feb 2022) [pdf]

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

See also:

• Wikipedia: Braid group

As an example of motion general “motion groups” (such as loop braid groups):

• Deborah L. Goldsmith, The theory of motion groups, Michigan Math. J. 28 1 (1981) 3-17 [doi:10.1307/mmj/1029002454]

On presentations of surface braid groups:

• Paolo Bellingeri: On presentation of Surface Braid Groups [arXiv:math/0110129]

Algebraic presentation of braid groups:

• Warren Dicks, Edward Formanek, around Ex. 15.2 of: Algebraic Mapping-Class Groups of Orientable Surfaces with Boundaries, in: Infinite Groups: Geometric, Combinatorial and Dynamical Aspects, Progress in Mathematics 248 Birkhäuser (2005) [doi :10.1007/3-7643-7447-0_4]

• Bacardit & Dicks 2009

See also:

• E. A. Gorin, V. Ya. Lin: Algebraic equations with continuous coefficients and some problems of the algebraic theory of braids, Mathematics of the USSR-Sbornik 7 4 (1969) 569–596 [doi:10.1070/SM1969v007n04ABEH001104, mathnet:sm3572, pdf]

More finite 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]

On the framed braid group:

• Ki Hyoung Ko, Lawrence Smolinsky: The framed braid group and 3-manifolds, Proc. Amer. Math. Soc. 115 (1992) 541-551 [doi:10.1090/S0002-9939-1992-1126197-1, pdf]

• Ki Hyoung Ko, Lawrence Smolinsky: The framed braid group and representations, in: Knots 90, Proceedings in Mathematics, De Gruyter (1992) 289-297 [doi:10.1515/9783110875911, pdf]

• R. Krasauskas: Crossed simplicial groups of framed braids and mapping class groups of surfaces, Lith Math J 36 (1996) 263–281 [doi:10.1007/BF02986853]

• Bellingeri & Gervais 2012

• Akishi Ikeda: Homological and Monodromy Representations of Framed Braid Groups, Commun. Math. Phys. 359 (2018) 1091–1121 [doi:10.1007/s00220-017-3036-1, arXiv:1702.03918]

• Marco De Renzi, Azat M. Gainutdinov, Nathan Geer, Bertrand Patureau-Mirand, Ingo Runkel, §3.1 in: Mapping Class Group Representations From Non-Semisimple TQFTs, Commun. Contemp. Math. (2021) 2150091 [arXiv:2010.14852, doi:10.1142/S0219199721500917]

  (in the context of Reshetikhin-Turaev TQFT)

• Ying Hong Tham: §3.2.1 in: On the Category of Boundary Values in the Extended Crane-Yetter TQFT, PhD thesis, Stony Brook (2021) [arXiv:2108.13467]

  (in the context of the Crane-Yetter model)

• Iordanis Romaidis, pp 37 in: Mapping class group actions and their applications to 3D gravity, PhD thesis, Hamburg (2022) [ediss:9945]

  (in the context of Reshetikhin-Turaev TQFT)

• Iordanis Romaidis, Ingo Runkel, p. 8 of: CFT correlators and mapping class group averages [arXiv:2309.14000]

  (in the context of Reshetikhin-Turaev TQFT)

• Lukas Woike: The Cyclic and Modular Microcosm Principle in Quantum Topology [arXiv:2408.02644]

  (in relation to the framed little disk operad)

• Anastasios Kokkinakis: Framed Braid Equivalences [arXiv:2503.05342]

More on the relation of braid groups to mapping class groups:

• M. Amram, R. Lawrence, U. Vishne, Artin Covers of the Braid Group, Journal of Knot Theory and Its Ramifications 21 07 (2012) 1250061 [doi:10.1142/S0218216512500617, pdf]

• Dan Margalit, Rebecca R. Winarski, Braid groups and mapping class groups: The Birman–Hilden theory, Bull. London Math. Soc. 53 3 (2021) 643-659 [arXiv:1703.03448, doi:10.1112/blms.12456]

• Paolo Bellingeri, Sylvain Gervais: Surface framed braids, Geom Dedicata 159 (2012) 51–69 [doi:10.1007/s10711-011-9645-5, arXiv:1001.4471]

More on the braid representation on automorphisms of free groups:

• 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]

• Tetsuya Ito, Actions of the nn-strand braid groups on the free group of rank n which are similar to the Artin representation, Quart. J. Math 66 (2015) 563-581 [arXiv;1406.2411, doi:10.1093/qmath/hau033]

On the group homology and group cohomology of braid groups:

• Vladimir Vershinin, Homology of Braid Groups and their Generalizations, Banach Center Publications (1998) 42 1 421-446 (pdf, dml:208821)

Relation of automorphism groups of the profinite completion of braid groups to the Grothendieck-Teichmüller group:

• Pierre Lochak, Leila Schneps, The Grothendieck-Teichmüller group and automorphisms of braid groups, in: The Grothendieck Theory of Dessins d’Enfant, Cambridge University Press (1994, 2011) [pdf, doi:10.1017/CBO9780511569302]

On braid subgroups:

• Dale Rolfsen: Braid subgroup normalisers, commensurators and induced representations, Invent Math 130 (1997) 575–587 [doi:10.1007/s002220050194, pdf]

• D. B. McReynolds: The congruence subgroup problem for braid groups: Thurston’s proof, New York Jour. Math. 18 (2012) 925-942 [arXiv:0901.4663]

• Charalampos Stylianakis: Congruence subgroups of braid groups, International Journal of Algebra and Computation 28 02 (2018) 345-364 [doi:10.1142/S0218196718500169]

On orderings of the braid group:

• Patrick Dehornoy: Braid groups and left distributive operations, Transactions AMS 345 1 (1994) 115150 [doi:10.1090/S0002-9947-1994-1214782-4, pdf]

• H. Langmaack, Verbandstheoretische Einbettung von Klassen unwesentlich verschiedener Ableitungen in die Zopfgruppe , Computing 7 no.3-4 (1971) pp.293-310.

On geometric presentations of braid groups:

• Byung Hee An, Tomasz Maciazek, Geometric Presentations of Braid Groups for Particles on a Graph, Communications in Mathematical Physics 384 (2021) 1109-1140 [doi:10.1007/s00220-021-04095-x]

On G-structure for G=Br ∞G = Br_\infty the infinite braid group:

• 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, pdf]

Graph braid groups

• Daniel Farley, Lucas Sabalka, Presentations of Graph Braid Groups (arXiv:0907.2730)

• Ki Hyoung Ko, Hyo Won Park, Characteristics of graph braid groups (arXiv:1101.2648)

• Byung Hee An, Tomasz Maciazek, Geometric presentations of braid groups for particles on a graph (arXiv:2006.15256)

Relation to moduli space of monopoles

On moduli spaces of monopoles related to braid groups:

• Fred Cohen, Ralph Cohen, B. M. Mann, R. J. Milgram, The topology of rational functions and divisors of surfaces, Acta Math (1991) 166: 163 (doi:10.1007/BF02398886)

• Ralph Cohen, John D. S. Jones Monopoles, braid groups, and the Dirac operator, Comm. Math. Phys. Volume 158, Number 2 (1993), 241-266 (euclid:cmp/1104254240)