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

28 KNOTS
Figure 2.13. The product of two braids.
strands. This operation consists simply of placing the braids end to
end (by joining the upper part of the second braid to the lower part of
the first), as in Figure 2.13.
It turns out that the product of braids possesses several properties
that resemble the ordinary product of numbers. First, there is a braid
known as the unit braid (e), a braid that, like the number l, does not
change what it multipli es. This is the trivial braid, whose strands hang
vertically without crossing. Sure enough, appending a trivial braid to a
given braid amounts to extending its strands, which do es not change
the class of the braid in any way.
Second, for each braid b there exists an inverse braid, b- I, whose
product with b gives the trivial braid: b . b- I = e (just as fo r each
number n, its product with the inverse number n-I = lIn is equal to
one, n . n-I = 1). As can be seen in Figure 2.14, the inverse braid is the