Sossinsky:Knots. Mathematics with a twist.pdf - English

30 KNOTS
2 3 • n-1 n
I I I
b,_,
2 3 4
Figure 201 So Algebraic representation of a braido
senting a braid by a word-the algebraic encoding of that braido Indeed,
moving along a braid from top to bottom, we see that it is the
successive product of braids each with a singie crossing (Figure 2015);
we calI these elementary braids and denote them by bi, bl, o o o , bn-I (for
braids with n strands)o
So we have repiaced braids-geometric objects-by words: their algebraic
codeso But ree alI that the geometrie braids possess an equiva­
Ience reiation, nameIy, isotopyo What does that mean algebraicalIy?
Artin had an answer to this questiono He found a series of algebraic re­
Iations between braid words that gave an adequate algebraic description
of their isotopyo These reiations are commutativity for distant
braids
i, j = 1, 2, o o o , n - l
5
b,
b;
b3
b,
-,
b3
b4