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

18 KNOTS
is true for the more general case of Iinks, of which knots are just an exampIe.)
Alexander probably hoped that his theorem would be a decisive step
forward in classifying knots. Indeed, as we will see Iater, braids are
much simpier objects than knots; the set of braids possesses a very
clear algebraic structure that enabies one to classify them. Is it reasonabIe,
then, to try to use braids to classify knots? How did this idea develop?
We will see at the end of the chapter.
For now, Iet us return to Alexander's theorem: How can it be
proved? Given a knot, how does one find the braid whose closure
would be that knot? First of all, note that the desired braid can easily
be seen when the braid is rolled up in a coil, that is, when it aIways
turns in the same direction around a certain point (as the knot in Fig-
(a)
(b)
Figure 2.3. Closure of two braids.