Contents

group theory

# Contents

## 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

Let $C_n \hookrightarrow \mathbb{C}^n$ be the space of configurations of n points in the complex plane, whose elements are those n-tuples $(z_1, \ldots, z_n)$ such that $z_i \neq z_j$ whenever $i \neq j$. The symmetric group $S_n$ acts on $C_n$ by permuting coordinates. Let $C_n/S_n$ be the orbit space (the space of $n$-element subsets of $\mathbb{C}$ if one likes), and let $[z_1, \ldots, z_n]$ be the image of $(z_1, \ldots, z_n)$ under the quotient $\pi: C_n \to C_n/S_n$. We take $p = (1, 2, \ldots, n)$ as basepoint for $C_n$, and $[p] = [1, 2, \ldots n]$ as basepoint for $C_n/S_n$.

###### Definition

The braid group $Br_n$ is the fundamental group $\pi_1(C_n/S_n, [p])$. The pure braid group $P_n$ is $\pi_1(C_n, p)$.

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

$1 \to P_n \to Br_n \to S_n \to 1$

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

### Group-theoretic definition

#### Artin presentation

The Artin braid group, $Br_{n+1}$, defined using $n+1$ strands is a group given 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 $1 \leq i \lt n$.

#### In terms of automorphisms on free groups

The braid group $B_n$ 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

$B_n = MCG(X_n) \to Aut(F_n)$

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 $Braids_{2k}$ on $2 k$ strands:

$\Sigma^\infty \mathcal{M}_k \;\simeq\; \Sigma^\infty Braids_{2k}$

## Examples

The first few examples for low values of $n$:

###### The group $Br_1$

By default, $Br_1$ has no generators and no relations, so is trivial.

###### The group $Br_2$

By default, $Br_2$ has one generator and no relations, so is infinite cyclic.

###### The group $Br_3$

(We will simplify notation writing $u = y_1$, $v = y_2$.)

This then has presentation

$\mathcal{P} = ( u,v : r \equiv u v u v^{-1} u^{-1} v^{-1}).$

It is also the ‘trefoil group’, i.e., the fundamental group of the complement of a trefoil knot.

###### The group $Br_4$

Simplifying notation as before, we have generators $u,v,w$ and relations

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

## Surface braid groups

In terms of the geometric definition above, it is possible to consider configurations of points on surfaces other than the plane, which gives rise to the more general notion of a surface braid group. For example, the Hurwitz braid group (or sphere braid group) comes from considering configurations of points on 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:

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

## References

### General

Classical references are

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

• R. H. Fox, L. Neuwirth, The braid groups, Math. Scand. 10 (1962) pp.119-126. pdf, MR150755

Textbook accounts:

• 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)

On the group homology and group cohomology of braid groups:

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.

### 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):

Review:

in relation to modular tensor categories:

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

• 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)

### Relation to moduli space of monopoles

On moduli spaces of monopoles related to braid groups:

### 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”).

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:

#### 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

Last revised on May 23, 2021 at 04:10:18. See the history of this page for a list of all contributions to it.