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The theory of the braids is the study of the braids, mathematical object formalizing what is called braid (or plait) in the everyday life.

Definition

That is to say $A = \ left \ {a_1,…, a_n \ right \}$ a whole of N points of $\ mathbb {D}$ the open disc unit of $\ mathbb {C}$ .

One calls bit the graph of $b$ a continuous application of $I= \ left$ in the open disc unit of $\ mathbb {C}$ , whose ends $b (0)$ and $b (1)$ belong to $A$ .

One calls braid with $n$ bits the meeting of $n$ disjoined bits.

Reformulation

Geometrically one projects the representation 3D of a braid in the plan. A diagram of braid thus is obtained. In order not to lose information with respect to space in 3 dimensions it is necessary to indicate, when two bits cross, which passes in front of the other.

The order of arrival of the bits is different from the order departure. The positions underwent a transformation; here it is about the permutation of (1 4 3). The study of the braids is related to the study of the Permutation S and offers an additional data by adding an idea of way (order of the operations carried out in the transformation) non-existent in the permutations. With each diagram of braids of $n$ bits one associates a permutation of $\ {1, \ ldots, N \}$ and with each permutation of $\ {1, \ ldots, N \}$ one associates several diagrams of braid.

This leads us to introduce a particular braid, the commonplace braid where no crossing takes place between the various bits. For example, here the diagram of the commonplace braid with four bits.

One wants to provide the unit with the diagrams of braid of a mathematical structure. Let us note first of all that the length of the bits imports little in the structure of the braid; it is completely characterized by the crossings of the bits and the order in which these crossings are carried out. Thus even if they do not have the same length, two diagrams of braid which have the same crossings in the same order are regarded as equal.

That enables us to provide the unit with the diagrams of braid of a structure of monoid, in the following way: One defines the product of two diagrams of braid having the same number of bits $n$ by operating a concatenation, i.e by hanging the second at the end of the first what gives a new diagram of braid to $n$ bits:

This product is associative but noncommutative. One notices moreover than the product of a diagram of braid $\ beta$ and diagram of the commonplace braid gives a diagram of braid identical to the diagram $\ beta$ . Thus the commonplace diagram is a neutral element for the concatenation.

The whole of the diagrams of braid with $n$ bits provided with the concatenation is thus a Monoïde. Let us note the $T_n$ .

Notice

So that the diagrams of braids and their product correspond to the permutations and their composition it is necessary to read the diagrams of braids upwards. For example, the diagram of braids above has as a permutation

(1 4 3)

Thus let us take two diagrams of braids $b_1$ and $b_2$ of respective associated permutation $s_1$ and $s_2$ . The product $b_1b_2$ has as a permutation $s_1 \ circ s_2$ .

To study the braids, they should be compared with respect to their way and of their associated permutation. In a diagram of braids, certain crossings are independent from/to each other.

Two diagrams of braids are known as “isotopes” if one can obtain one from the other by moving the bits without “cutting them” and without touching at the ends.

The relation of isotopy on $T_n$ is a relation of equivalence.

Two diagrams of braids isotopes represent same the Permutation, but the reciprocal one is false: two diagrams having the same associated permutation are not necessarily isotopes.

Group braids with $n$ bits

By quotientant $T_n$ by the relation of isotopy one obtains a structure of group on the whole of the diagrams of braids with $n$ bits. One notes $B_n$ and one calls “group of braid with N bits” the group thus obtained. The neutral element being obviously the class of the commonplace diagram, the reverse of a diagram is the diagram obtained by taking its image mirror.

By simplification one calls braid with $n$ bits an element of $B_n$ .

Group braids

One plunges

B_n

B_ {n+1}

by transforming the braids to

n

bits into braids to

n+1

bits in the following way. One adds on the right a

n+1

ème bits which does not cross any other of it, as one sees it in the following example:

One notes $B_ {\ infty}$ the group

$B_ {\ infty} = \ bigcup_ {N \ geq 0} B_n$ .

Applications and generalizations

The group of braids like fundamental groups

The group of braid with N bits is isomorphous with the fundamental group of the space of configuration

\ left \ {(z_1, \ dowries, z_n) \ in \ mathbb {C} ^n \ |\ \ forall I \ neq J \ in \ {1 \ dowries N \}, \ z_i \ neq z_j \ right \}

See too

Theory of the nodes

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