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Our interaction model is based on the clever but simple geometric construction named the Reidemeister moves, which allows 3D topological manipulations using rather simple 2D moves. On the other hand, two such connections which can be made to look the same by "pulling the strands" are considered the same braid:.
All strands are required to move from left to right; knots like the following are not considered braids:. Any two braids can be composed by drawing the first next to the second, identifying the four items in the middle, and connecting corresponding strands:. The above composition of braids is indeed a group operation. The identity element is the braid consisting of four parallel horizontal strands, and the inverse of a braid consists of that braid which "undoes" whatever the first braid did, which is obtained by flipping a diagram such as the ones above across a vertical line going through its centre.
The first two example braids above are inverses of each other. Braid theory has recently been applied to fluid mechanics , specifically to the field of chaotic mixing in fluid flows. To put the above informal discussion of braid groups on firm ground, one needs to use the homotopy concept of algebraic topology , defining braid groups as fundamental groups of a configuration space.
Alternatively, one can define the braid group purely algebraically via the braid relations, keeping the pictures in mind only to guide the intuition. When X is the plane, the braid can be closed , i. 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. 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.
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In , Andrey Markov Jr. 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. 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. Braid groups were introduced explicitly by Emil Artin in , although as Wilhelm Magnus pointed out in  they were already implicit in Adolf Hurwitz 's work on monodromy from Braid groups may be described by explicit presentations , as was shown by Emil Artin in As Magnus says, Hurwitz gave the interpretation of a braid group as the fundamental group of a configuration space cf.
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.
Book A Study Of Braids Mathematics And Its Applications 1999
By forgetting how the strands twist and cross, every braid on n strands determines a permutation on n elements. 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:.
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. This sequence splits and therefore pure braid groups are realized as iterated semi-direct products of free groups. Here is a construction of this isomorphism. Alternately, one common presentation for the modular group is.
The center of B 3 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.
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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. 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.
A Study of Braids, Mathematics and Its Applications by Kunio Murasugi | | Booktopia
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. 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. In analogy with the action of the symmetric group by permutations, in various mathematical settings there exists a natural action of the braid group on n -tuples of objects or on the n -folded tensor product that involves some "twists".
Consider an arbitrary group G and let X be the set of all n -tuples of elements of G whose product is the identity element of G.
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