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.
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 $C_n/S_n$.
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
which is part of a long exact homotopy sequence corresponding to the fibration $\pi: C_n \to C_n/S_n$.
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 $1 \leq i \lt n$.
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
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
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
(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:
The first few examples for low values of $n$:
By default, $Br_1$ has no generators and no relations, so is trivial.
By default, $Br_2$ has one generator and no relations, so is infinite cyclic.
(We will simplify notation writing $u = y_1$, $v = y_2$.)
This then has presentation
It is also the ‘trefoil group’, i.e., the fundamental group of the complement of a trefoil knot.
Simplifying notation as before, we have generators $u,v,w$ and relations
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:
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 |
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
A recent monograph is
See also
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 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)
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:
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
A followup proposal was to use the problem of reversing E-multiplication in braid groups, thought to remedy the previous problems.
Original article:
Review:
But other problems were found with this approach, rendering it non-viable.
Original article:
Review:
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)
Last revised on April 22, 2021 at 02:51:23. See the history of this page for a list of all contributions to it.