Braids and Braids Group

Topological Braids

Braid

Fix a collection of  points  in the unit disk, a braid on strands is a smooth embedding such that and for some permutation .

Braid Equivalence

Two braids are equivalent if they are ambient isotopic in .

Braid Stacking

Braid Group

The Braid Group

Set of all braids up to equivalence forms a group under the operation of stacking braids.

Artin's Theorem: The Braid Relation

The braid group is generated by the set of -braids with the following relations:

e.g. Consider the braids on strands, there are generators and their inverses:

Proposition

.

Proof The braid group on strands is generated by without any relation. Thus .

Complex Polynomial Interpretation

The Word Problem

The Word Problem

The word problem for a group presented by a system of generators and relations, in particular for the braid group, is to find an algorithm that determines, for any pair of words in the generators of the group, whether or not they present the same element (i.e., whether or not one can be taken to the other by applying the group’s relations and the trivial relations). The word problem (which of course is always solvable for commutative groups) is solvable for the braid group for any .

Fully Reduced Word

A word is said to be completely reduced if, for any integer , any occurrence of the letter is separated from any occurrence of the letter by at least one occurrence of a letter with .

e.g. is fully reduced, is not.

Proposition

Any element can be represented by a fully reduced word of generators.

Proof Suppose generates . And represents some . If is not reduced, then it is of the form with , is completely reduced, and contains only with .

The Dehornoy Algorithm

Given two words of the generators of , apply elementary reductions to both words until they are completely reduced. Then compare the two completely reduced words to see if they are equal.