Braid group

When X is the plane, the braid can be closed, i.e., corresponding ends can be connected in pairs, to form a link, i.e., a possibly intertwined union of possibly knotted loops in three dimensions. The number of components of the link can be anything from 1 to n, depending on the permutation of strands determined by the link. A theorem of J. W. Alexander demonstrates that every link can be obtained in this way as the "closure" of a braid. Compare with string links.

Different braids can give rise to the same link, just as different crossing diagrams can give rise to the same knot. In 1935, Andrey Markov Jr. described two moves on braid diagrams that yield equivalence in the corresponding closed braids.[6] A single-move version of Markov's theorem, was published by in 1997.[7]

Vaughan Jones originally defined his polynomial as a braid invariant and then showed that it depended only on the class of the closed braid.

The Markov theorem gives necessary and sufficient conditions under which the closures of two braids are equivalent links.[8]

The "braid index" is the least number of strings needed to make a closed braid representation of a link. It is equal to the least number of Seifert circles in any projection of a knot.[9]

Consider the following three braids:

${\displaystyle \sigma _{1}}$	${\displaystyle \sigma _{2}}$	${\displaystyle \sigma _{3}}$

Every braid in ${\displaystyle B_{4}}$ can be written as a composition of a number of these braids and their inverses. In other words, these three braids generate the group ${\displaystyle B_{4}}$. To see this, an arbitrary braid is scanned from left to right for crossings; beginning at the top, whenever a crossing of strands ${\displaystyle i}$ and ${\displaystyle i+1}$ is encountered, ${\displaystyle \sigma _{i}}$ or ${\displaystyle \sigma _{i}^{-1}}$ is written down, depending on whether strand ${\displaystyle i}$ moves under or over strand ${\displaystyle i+1}$. Upon reaching the right end, the braid has been written as a product of the ${\displaystyle \sigma }$'s and their inverses.

It is clear that

(i) ${\displaystyle \sigma _{1}\sigma _{3}=\sigma _{3}\sigma _{1}}$,

while the following two relations are not quite as obvious:

(iia) ${\displaystyle \sigma _{1}\sigma _{2}\sigma _{1}=\sigma _{2}\sigma _{1}\sigma _{2}}$,

(iib) ${\displaystyle \sigma _{2}\sigma _{3}\sigma _{2}=\sigma _{3}\sigma _{2}\sigma _{3}}$

(these relations can be appreciated best by drawing the braid on a piece of paper). It can be shown that all other relations among the braids ${\displaystyle \sigma _{1}}$, ${\displaystyle \sigma _{2}}$ and ${\displaystyle \sigma _{3}}$ already follow from these relations and the group axioms.

Generalising this example to ${\displaystyle n}$ strands, the group ${\displaystyle B_{n}}$ can be abstractly defined via the following presentation:

${\displaystyle B_{n}=\left\langle \sigma _{1},\ldots ,\sigma _{n-1}\mid \sigma _{i}\sigma _{i+1}\sigma _{i}=\sigma _{i+1}\sigma _{i}\sigma _{i+1},\sigma _{i}\sigma _{j}=\sigma _{j}\sigma _{i}\right\rangle ,}$

where in the first group of relations ${\displaystyle 1\leq i\leq n-2}$ and in the second group of relations, ${\displaystyle i-j\geq 2}$. This presentation leads to generalisations of braid groups called Artin groups. The cubic relations, known as the braid relations, play an important role in the theory of Yang–Baxter equations.

• The braid group ${\displaystyle B_{1}}$ is trivial, ${\displaystyle B_{2}}$ is the infinite cyclic group ${\displaystyle \mathbb {Z} }$, and ${\displaystyle B_{3}}$ is isomorphic to the knot group of the trefoil knot – in particular, it is an infinite non-abelian group.
• The n-strand braid group ${\displaystyle B_{n}}$ embeds as a subgroup into the ${\displaystyle (n+1)}$-strand braid group ${\displaystyle B_{n+1}}$ by adding an extra strand that does not cross any of the first n strands. The increasing union of the braid groups with all ${\displaystyle n\geq 1}$ is the infinite braid group ${\displaystyle B_{\infty }}$.
• All non-identity elements of ${\displaystyle B_{n}}$ have infinite order; i.e., ${\displaystyle B_{n}}$ is torsion-free.
• There is a left-invariant linear order on ${\displaystyle B_{n}}$ called the Dehornoy order.
• For ${\displaystyle n\geq 3}$, ${\displaystyle B_{n}}$ contains a subgroup isomorphic to the free group on two generators.
• There is a homomorphism ${\displaystyle B_{n}\to \mathbb {Z} }$ defined by σ_i ↦ 1. So for instance, the braid σ₂σ₃σ₁⁻¹σ₂σ₃ is mapped to 1 + 1 − 1 + 1 + 1 = 3. This map corresponds to the abelianization of the braid group. Since σ_i^k ↦ k, then σ_i^k is the identity if and only if ${\displaystyle k=0}$. This proves that the generators have infinite order.

By forgetting how the strands twist and cross, every braid on n strands determines a permutation on n elements. This assignment is onto and compatible with composition, and therefore becomes a surjective group homomorphism B_n → S_n from the braid group onto the symmetric group. The image of the braid σ_i ∈ B_n is the transposition s_i = (i, i+1) ∈ S_n. These transpositions generate the symmetric group, satisfy the braid group relations, and have order 2. This transforms the Artin presentation of the braid group into the Coxeter presentation of the symmetric group:

${\displaystyle S_{n}=\left\langle s_{1},\ldots ,s_{n-1}|s_{i}s_{i+1}s_{i}=s_{i+1}s_{i}s_{i+1},s_{i}s_{j}=s_{j}s_{i}{\text{ for }}|i-j|\geq 2,s_{i}^{2}=1\right\rangle .}$

The kernel of the homomorphism B_n → S_n is the subgroup of B_n called the pure braid group on n strands and denoted P_n. This can be seen as the fundamental group of the space of n-tuples of distinct points of the euclidian plane. In a pure braid, the beginning and the end of each strand are in the same position. Pure braid groups fit into a short exact sequence

${\displaystyle 1\to F_{n-1}\to P_{n}\to P_{n-1}\to 1.}$

This sequence splits and therefore pure braid groups are realized as iterated semi-direct products of free groups.

${\displaystyle B_{3}}$ is the universal central extension of the modular group.

The braid group ${\displaystyle B_{3}}$ is the universal central extension of the modular group ${\displaystyle \mathrm {PSL} (2,\mathbb {Z} )}$, with these sitting as lattices inside the (topological) universal covering group

${\displaystyle {\overline {\mathrm {SL} (2,\mathbb {R} )}}\to \mathrm {PSL} (2,\mathbb {R} )}$.

Furthermore, the modular group has trivial center, and thus the modular group is isomorphic to the quotient group of ${\displaystyle B_{3}}$ modulo its center, ${\displaystyle Z(B_{3}),}$ and equivalently, to the group of inner automorphisms of ${\displaystyle B_{3}}$.

Here is a construction of this isomorphism. Define

${\displaystyle a=\sigma _{1}\sigma _{2}\sigma _{1},\quad b=\sigma _{1}\sigma _{2}}$.

From the braid relations it follows that ${\displaystyle a^{2}=b^{3}}$. Denoting this latter product as ${\displaystyle c}$, one may verify from the braid relations that

${\displaystyle \sigma _{1}c\sigma _{1}^{-1}=\sigma _{2}c\sigma _{2}^{-1}=c}$

implying that ${\displaystyle c}$ is in the center of ${\displaystyle B_{3}}$. Let ${\displaystyle C}$ denote the subgroup of ${\displaystyle B_{3}}$ generated by c, since C ⊂ Z(B₃), it is a normal subgroup and one may take the quotient group B₃/C. We claim B₃/C ≅ PSL(2, Z); this isomorphism can be given an explicit form. The cosets σ₁C and σ₂C map to

${\displaystyle \sigma _{1}C\mapsto R={\begin{bmatrix}1&1\\0&1\end{bmatrix}}\qquad \sigma _{2}C\mapsto L^{-1}={\begin{bmatrix}1&0\\-1&1\end{bmatrix}}}$

where L and R are the standard left and right moves on the Stern–Brocot tree; it is well known that these moves generate the modular group.

Alternately, one common presentation for the modular group is

${\displaystyle \langle v,p\,|\,v^{2}=p^{3}=1\rangle }$

where

${\displaystyle v={\begin{bmatrix}0&1\\-1&0\end{bmatrix}},\qquad p={\begin{bmatrix}0&1\\-1&1\end{bmatrix}}.}$

Mapping a to v and b to p yields a surjective group homomorphism B₃ → PSL(2, Z).

The center of B₃ is equal to C, a consequence of the facts that c is in the center, the modular group has trivial center, and the above surjective homomorphism has kernel C.

The braid group B_n can be shown to be isomorphic to the mapping class group of a punctured disk with n punctures. This is most easily visualized by imagining each puncture as being connected by a string to the boundary of the disk; each mapping homomorphism that permutes two of the punctures can then be seen to be a homotopy of the strings, that is, a braiding of these strings.

Via this mapping class group interpretation of braids, each braid may be classified as periodic, reducible or pseudo-Anosov.

If a braid is given and one connects the first left-hand item to the first right-hand item using a new string, the second left-hand item to the second right-hand item etc. (without creating any braids in the new strings), one obtains a link, and sometimes a knot. Alexander's theorem in braid theory states that the converse is true as well: every knot and every link arises in this fashion from at least one braid; such a braid can be obtained by cutting the link. Since braids can be concretely given as words in the generators σ_i, this is often the preferred method of entering knots into computer programs.

The word problem for the braid relations is efficiently solvable and there exists a normal form for elements of B_n in terms of the generators σ₁, ..., σ_n−1. (In essence, computing the normal form of a braid is the algebraic analogue of "pulling the strands" as illustrated in our second set of images above.) The free GAP computer algebra system can carry out computations in B_n if the elements are given in terms of these generators. There is also a package called CHEVIE for GAP3 with special support for braid groups. The word problem is also efficiently solved via the Lawrence–Krammer representation.

In addition to the word problem, there are several known hard computational problems that could implement braid groups, applications in cryptography have been suggested.[13]

Elements of the braid group B_n can be represented more concretely by matrices. One classical such representation is Burau representation, where the matrix entries are single variable Laurent polynomials. It had been a long-standing question whether Burau representation was faithful, but the answer turned out to be negative for n ≥ 5. More generally, it was a major open problem whether braid groups were linear. In 1990, Ruth Lawrence described a family of more general "Lawrence representations" depending on several parameters. In 1996, Chetan Nayak and Frank Wilczek posited that in analogy to projective representations of SO(3), the projective representations of the braid group have a physical meaning for certain quasiparticles in the fractional quantum hall effect.[14] Around 2001 Stephen Bigelow and Daan Krammer independently proved that all braid groups are linear. Their work used the Lawrence–Krammer representation of dimension ${\displaystyle n(n-1)/2}$ depending on the variables q and t. By suitably specializing these variables, the braid group ${\displaystyle B_{n}}$ may be realized as a subgroup of the general linear group over the complex numbers.