Braid theory

An example of a spit with three arcs.

The braid theory is a section of topology and algebra that studies braids and braid groups composed of their equivalence classes.

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Spit Definition

Spit of $n$ ${\ displaystyle n}$ threads - an object consisting of two parallel planes $P_{0}$ ${\ displaystyle P_ {0}}$ and $P_{1}$ ${\ displaystyle P_ {1}}$ in three-dimensional space $\mathbb {R} ^{3}$ ${\ displaystyle \ mathbb {R} ^ {3}}$ containing ordered sets of points $a_{1},a_{2},\dots ,a_{n}\in P_{0}$ ${\ displaystyle a_ {1}, a_ {2}, \ dots, a_ {n} \ in P_ {0}}$ and $b_{1},b_{2},\dots ,b_{n}\in P_{1}$ ${\ displaystyle b_ {1}, b_ {2}, \ dots, b_ {n} \ in P_ {1}}$ and from $n$ ${\ displaystyle n}$ disjoint simple arcs $l_{1},l_{2},\dots ,l_{n}$ ${\ displaystyle l_ {1}, l_ {2}, \ dots, l_ {n}}$ crossing each parallel plane $P_{t}$ ${\ displaystyle P_ {t}}$ between $P_{0}$ ${\ displaystyle P_ {0}}$ and $P_{1}$ ${\ displaystyle P_ {1}}$ one time and connecting points $\{a_{i}\}$ ${\ displaystyle \ {a_ {i} \}}$ with dots $\{b_{i}\}$ ${\ displaystyle \ {b_ {i} \}}$ .

It is usually considered that the points $a_{1},a_{2},\dots ,a_{n}$ ${\ displaystyle a_ {1}, a_ {2}, \ dots, a_ {n}}$ lie on a straight line $l_{0}$ ${\ displaystyle l_ {0}}$ at $P_{0}$ ${\ displaystyle P_ {0}}$ and dots $b_{1},b_{2},\dots ,b_{n}$ ${\ displaystyle b_ {1}, b_ {2}, \ dots, b_ {n}}$ on the straight $l_{1}$ ${\ displaystyle l_ {1}}$ at $P_{1}$ ${\ displaystyle P_ {1}}$ parallel $l_{0}$ ${\ displaystyle l_ {0}}$ , and $a_{i}$ ${\ displaystyle a_ {i}}$ located under $b_{i}$ ${\ displaystyle b_ {i}}$ for each $i$ ${\ displaystyle i}$ .

The braids are depicted in projection on the plane passing through $l_{0}$ ${\ displaystyle l_ {0}}$ and $l_{1}$ ${\ displaystyle l_ {1}}$ This projection can be brought into general position so that there are only a finite number of double points lying in pairs in different levels, and the intersections are transversal .

Braid Group

In the set of all braids with n threads and with fixed $P_{0},P_{1},\{a_{i}\},\{b_{i}\}$ ${\ displaystyle P_ {0}, P_ {1}, \ {a_ {i} \}, \ {b_ {i} \}}$ an equivalence relation is introduced. It is determined by homeomorphisms. $h:\Pi \to \Pi$ ${\ displaystyle h: \ Pi \ to \ Pi}$ where $\Pi$ ${\ displaystyle \ Pi}$ - area between $P_{0}$ ${\ displaystyle P_ {0}}$ and $P_{1}$ ${\ displaystyle P_ {1}}$ identical with $P_{0}\cup P_{1}$ ${\ displaystyle P_ {0} \ cup P_ {1}}$ . Braids $\alpha$ ${\ displaystyle \ alpha}$ and $\beta$ ${\ displaystyle \ beta}$ are equivalent if there is such a homeomorphism $h$ ${\ displaystyle h}$ , what $h(\alpha )=\beta$ ${\ displaystyle h (\ alpha) = \ beta}$ .

Equivalence classes, hereinafter also referred to as braids, form a braid group $B(n)$ ${\ displaystyle B (n)}$ . Unit braid is an equivalence class containing a braid of n parallel segments. Spit $\alpha ^{-1}$ ${\ displaystyle \ alpha ^ {- 1}}$ backward spit $\alpha$ ${\ displaystyle \ alpha}$ determined by the reflection in the plane $P_{1/2}$ ${\ displaystyle P_ {1/2}}$

Spit thread connects $a_{i}$ ${\ displaystyle a_ {i}}$ with $b_{j_{i}}$ ${\ displaystyle b_ {j_ {i}}}$ and defines a substitution, an element of the symmetric group $S_{n}$ ${\ displaystyle S_ {n}}$ . If this permutation is identical, then the braid is called a colored (or pure) braid. This mapping defines an epimorphism. $B(n)$ ${\ displaystyle B (n)}$ per group $S_{n}$ ${\ displaystyle S_ {n}}$ permutations of n elements whose core is a subgroup $K(n)$ ${\ displaystyle K (n)}$ corresponding to all pure braids, so there is a short exact sequence

0\to K(n)\to B(n)\to S_{n}\to 0

{\ displaystyle 0 \ to K (n) \ to B (n) \ to S_ {n} \ to 0}

Literature

Sosinsky A., Braids and knots. (inaccessible link) Quantum No. 2, 1989, p. 6-14

Braid theory

Content

Spit Definition

Braid Group

See also

Literature

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