nLab braid group

Idea

The braid group $Br(n)$ is the group whose elements are isotopy classes of $n$ 1-dimensional braids running vertically in 3-dimensional Cartesian space, the group operation being their concatenation.

Here a braid with $n$ strands is thought of as $n$ pieces of string joining $n$ points at the top of the diagram with $n$-points at the bottom.

(This is a picture of a 3-strand braid.)

We can transform / ‘isotope’ these braid diagrams just as we can transform knot diagrams, again using Reidemeister moves. The ‘isotopy’ classes of braid diagrams form a group in which the composition is obtained by putting one diagram above another.

The identity consists of $n$ vertical strings, so the inverse is obtained by turning a diagram upside down:

This is the inverse of the first 3-braid we saw.

There are useful group presentations of the braid groups. We will return later to the interpretation of the generators and relations in terms of diagrams.

Definition

Geometric definition

(due to Fadell & Neuwirth 1962, p. 118, Fox & Neuwirth 1962, §7, reviewed in Williams 2020, pp. 9)

Ordinary braid group

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

We say this in more detail:

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

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

The symmetric group $S_n$ acts on $C_n$ by permuting coordinates. Let:

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

• $[z_1, \ldots, z_n]$ denote the image of $(z_1, \ldots, z_n)$ under the quotient coprojection $\pi \colon C_n \to C_n/S_n$ (i.e. its the equivalence class).

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

Definition

The braid group is the fundamental group of the configuration space of n unordered points:

$$Br(n) \;\coloneqq\; \pi_1 \big( C_n/S_n, [p] \big)$$

The pure braid group is the fundamental group of the configuration space of n ordered points:

$$PBr(n) \;\coloneqq\; \pi_1 \big( C_n, p \big) \,.$$

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

$$1 \to PBr(n) \to Br(n) \to Sym(n) \to 1$$

which is part of a long exact homotopy sequence corresponding to the fibration $\pi \colon C_n \to C_n/S_n$.

For general topological base spaces

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

For $\Sigma$ any surface, the fundamental group of the (ordered) configuration space of points in $\Sigma$ may be regarded as generalized (pure) braid group. 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 $X$.

For example for $X$ 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

Group-theoretic definition

Definition

(Artin presentation)
The Artin braid group, $Br({n+1})$, on $n+1$ strands is the finitely generated group given via generators and relations by:

• generators: $y_i$, $i = 1, \ldots, n$;

• relations:

  • $r_{i,j} \equiv y_i y_j y_i^{-1} y_j^{-1}$ for $i+1 \lt j$

  • $r_{i,i+1}\equiv y_i y_{i+1} y_i y_{i+1}^{-1} y_i^{-1} y_{i+1}^{-1}$ for $1 \leq i \lt n$.

(e.g. Fox & Neuwirth 1962, §7)

In terms of automorphisms on free groups

The braid group $Br(n)$ may be alternatively described as the mapping class group of a 2-disk $D^2$ with $n$ punctures (call it $X_n$). Meanwhile, the fundamental group $\pi_1(X_n)$ (with basepoint on the boundary) is a free group $F_n$ on $n$ generators; the functoriality of $\pi_1$ implies we have an induced homomorphism

$$Aut(X_n) \to Aut(\pi_1(X_n)) = Aut(F_n).$$

If an automorphism $\phi: X_n \to X_n$ is isotopic to the identity, then of course $\pi_1(\phi)$ is trivial, and so the homomorphism factors through the quotient $MCG(X_n) = Aut(X_n)/Aut_0(X)$, so we get a homomorphism

$$Br(n) = MCG(X_n) \to Aut(F_n)$$

and this turns out to be an injection.

Explicitly, the generator $y_i$ used in the Artin presentation above is mapped to the automorphism $\sigma_i$ on the free group on $n$ generators $x_1, \ldots, x_n$ defined by

$$\sigma_i(x_i) = x_{i+1}, \sigma_i(x_{i+1}) = x_{i+1}^{-1} x_i x_{i+1}, \; \else\; \sigma(x_j) = x_j.$$

Properties

Relation to moduli space of monopoles

Proposition

(moduli space of monopoles is stably weak homotopy equivalent to classifying space of braid group)

For $k \in \mathbb{N}$ there is a stable weak homotopy equivalence between the moduli space of k monopoles (?) and the classifying space of the braid group $Br({2k})$ on $2 k$ strands:

$$\Sigma^\infty \mathcal{M}_k \;\simeq\; \Sigma^\infty Br({2k})$$

(Cohen-Cohen-Mann-Milgram 91)

Examples

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

Example

The group $Br(1)$ has no generators and no relations, so is the trivial group:

$$Br(1) \;\simeq\; 1 \,.$$

Example

The group $Br(2)$ has one generator and no relations, so is the infinite cyclic group of integers:

$$Br(2) \;\simeq\; \mathbb{Z} \,.$$

Example

The group $Br(3)$ (we will simplify notation writing $u = y_1$, $v = y_2$) has presentation

$$Br(3) \;\simeq\; \mathcal{P} \coloneqq \big( u,v : r \equiv u v u v^{-1} u^{-1} v^{-1} \big).$$

This is also known as the “trefoil knot group”, i.e., the fundamental group of the complement of a trefoil knot.

Example

The group $Br(4)$ (simplifying notation as before) has generators $u,v,w$ and relations:

• $r_u \equiv v w v w^{-1} v^{-1} w^{-1}$,
• $r_v \equiv u w u^{-1} w^{-1}$,
• $r_w \equiv u v u v^{-1} u^{-1} v^{-1}$.

Example

The Hurwitz braid group (or sphere braid group) is the surface braid group for $\Sigma$ the 2-sphere $S^2$. Algebraically, the Hurwitz braid group $H_{n+1}$ has all of the generators and relations of the Artin braid group $Br({n+1})$, plus one additional relation:

$$y_1 y_2 \dots y_{n-1} y_n^2 y_{n-1}\dots y_2 y_1$$

Related entries

• braid representation

• braid group statistics

  anyons, quantum Hall effect

  topological quantum computation

• braid group cryptography

• braid category

• infinitesimal braid relation

• loop braid group

chord diagrams	weight systems
linear chord diagrams, round chord diagrams Jacobi diagrams, Sullivan chord diagrams	Lie algebra weight systems, stringy weight system, Rozansky-Witten weight systems

knots	braids
chord diagram, Jacobi diagram	horizontal chord diagram
1T&4T relation	2T&4T relation/ infinitesimal braid relations
weight system	horizontal weight system
Vassiliev knot invariant	Vassiliev braid invariant
weight systems are associated graded of Vassiliev invariants	horizontal weight systems are cohomology of loop space of configuration space

References

General

Classical references:

• Joan S. Birman, Braids, links, and mapping class groups, Princeton Univ Press, 1974.

• 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$]$

Textbook accounts:

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

See also:

• Wikipedia: Braid group

• 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)

• Dale Rolfsen, New developments in the theory of Artin’s braid groups (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)

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)

For orderings of the braid group see

• Patrick Dehornoy, Braid groups and left distributive operations , Transactions AMS 345 no.1 (1994) pp.115–150.

• 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:

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

Braid group representations (as topological quantum gates)

On linear representations of braid groups (see also at braid group statistics and interpretation as quantum gates in topological quantum computation):

• Ivan Marin, On the representation theory of braid groups, Annales mathématiques Blaise Pascal, 20 2 (2013) 193-260 (arXiv:math/0502118, dml:275607)

Review:

• Camilo Arias Abad, Introduction to representations of braid groups, Rev. colomb. mat. vol.49 no.1 (2015) (arXiv:1404.0724, doi:10.15446/recolma.v49n1.54160)

• Toshitake Kohno, Introduction to representation theory of braid groups, Peking 2018 (pdf, pdf)

in relation to modular tensor categories:

• Colleen Delaney, Lecture notes on modular tensor categories and braid group representations, 2019 (pdf, pdf)

Braid representations seen inside the topological K-theory of the braid group‘s classifying space:

• Alejandro Adem, Daniel C. Cohen, Frederick R. Cohen, On representations and K-theory of the braid groups, Math. Ann. 326 (2003) 515-542 (arXiv:math/0110138, doi:10.1007/s00208-003-0435-8)

• Frederick R. Cohen, Section 3 of: On braid groups, homotopy groups, and modular forms, in: J.M. Bryden (ed.), Advances in Topological Quantum Field Theory, Kluwer 2004, 275–288 (pdf)

See also:

• R. B. Zhang, Braid group representations arising from quantum supergroups with arbitrary $q$ and link polynomials, Journal of Mathematical Physics 33, 3918 (1992) (doi:10.1063/1.529840)

As quantum gates for topological quantum computation with anyons:

• Louis H. Kauffman, Samuel J. Lomonaco, Braiding Operators are Universal Quantum Gates, New Journal of Physics, Volume 6, January 2004 (arXiv:quant-ph/0401090, doi:10.1088/1367-2630/6/1/134)

• Samuel J. Lomonaco, Louis Kauffman, Topological Quantum Computing and the Jones Polynomial, Proc. SPIE 6244, Quantum Information and Computation IV, 62440Z (2006) (arXiv:quant-ph/0605004)

  (braid group representation serving as a topological quantum gate to compute the Jones polynomial)

• Louis H. Kauffman, Samuel J. Lomonaco, Topological quantum computing and $SU(2)$ braid group representations, Proceedings Volume 6976, Quantum Information and Computation VI; 69760M (2008) (doi:10.1117/12.778068, rg:228451452)

• C.-L. Ho, A.I. Solomon, C.-H.Oh, Quantum entanglement, unitary braid representation and Temperley-Lieb algebra, EPL 92 (2010) 30002 (arXiv:1011.6229)

• Louis H. Kauffman, Majorana Fermions and Representations of the Braid Group, International Journal of Modern Physics AVol. 33, No. 23, 1830023 (2018) (arXiv:1710.04650, doi:10.1142/S0217751X18300235)

Introduction and review:

• Colleen Delaney, Eric C. Rowell, Zhenghan Wang, Local unitary representations of the braid group and their applications to quantum computing, Revista Colombiana de Matemáticas(2017), 50 (2):211 (arXiv:1604.06429, doi:10.15446/recolma.v50n2.62211)

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)

Braid group cryptography

Partly motivated by the possibility of quantum computation eventually breaking the security of cryptography based on abelian groups, such as elliptic curves, there are proposals to use non-abelian braid groups for purposes of cryptography (“post-quantum cryptography”).

Via Conjugacy Search

An early proposal was to use the Conjugacy Search Problem in braid groups as a computationally hard problem for cryptography. This approach, though, was eventually found not to be viable.

Original articles:

• Iris Anshel, M. Anshel and D. Goldfeld, An algebraic method for public-keycryptography, Math. Research Letters 6 (1999), 287–291 (pdf)

• K.H. Ko, S.J. Lee, J.H. Cheon , J.W. Han, J. Kang, C. Park , New Public-Key Cryptosystem Using Braid Groups, In: M. Bellare (ed.) Advances in Cryptology — CRYPTO 2000 Lecture Notes in Computer Science, vol 1880. Springer 2000 (doi:10.1007/3-540-44598-6_10)

Review:

• Karl Mahlburg, An Overview of Braid Group Encryption, 2004 (pdf)

• Parvez Anandam, Introduction to Braid Group Cryptography, 2006 (pdf)

• David Garber, Braid Group Cryptography, Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore (arXiv:0711.3941, doi:10.1142/9789814291415_0006)

• Cryptowiki, Cryptosystems based on braid groups

Via E-multiplication

A followup proposal was to use the problem of reversing E-multiplication in braid groups, thought to remedy the previous problems.

Original article:

• Iris Anshel, Derek Atkins, Dorian Goldfeld and Paul E Gunnells, WalnutDSA(TM): A Quantum-Resistant Digital Signature Algorithm (eprint:2017/058)

Review:

• Magnus Ringerud, WalnutDSA: Another attempt at braidgroup cryptography, 2019 (pdf)

But other problems were found with this approach, rendering it non-viable.

Original article:

• Matvei Kotov, Anton Menshov, Alexander Ushakov, An attack on the Walnut digital signature algorithm, Designs, Codes and Cryptography volume 87, pages 2231–2250 (2019) (doi:10.1007/s10623-019-00615-y)

Review:

• José Ignacio Escribano Pablos, María Isabel González Vasco, Misael Enrique Marriaga and Ángel Luis Pérez del Pozo, The Cracking of WalnutDSA: A Survey, in: Interactions between Group Theory, Symmetry and Cryptology, Symmetry 2019, 11(9), 1072 (doi:10.3390/sym11091072)

Further developments

The basic idea is still felt to be promising:

• Xiaoming Chen, Weiqing You, Meng Jiao, Kejun Zhang, Shuang Qing, Zhiqiang Wang, A New Cryptosystem Based on Positive Braids (arXiv:1910.04346)

• Garry P. Dacillo, Ronnel R. Atole, Braided Ribbon Group $C_n$-based Asymmetric Cryptography, Solid State Technology Vol. 63 No. 2s (2020) (JSST:5573)

But further attacks are being discussed:

• James Hughes, Allen Tannenbaum, Length-Based Attacks for Certain Group Based Encryption Rewriting Systems (arXiv:cs/0306032)

As are further ways around these:

• Xiaoming Chen, Weiqing You, Meng Jiao, Kejun Zhang, Shuang Qing, Zhiqiang Wang, A New Cryptosystem Based on Positive Braids (arXiv:1910.04346)

category: knot theory