Page 250 - J. C. Turner "History and Science of Knots"

A History of Topological Knot Theory 241

The braid group elements are open patches of weaving. They can be
closed by linking up the respective ends. The most obvious way this can be
done for an arbitrary ,0 E Bn is by pairing the ends at the ceiling one to one
with the corresponding ends at the floor. This yields the closed braid denoted
by 8.
The main evidence that braids would be useful in studying knots and links
was the existence of an (n-1)-dimensional representation of Bn, discovered by
Werner Burau in 1936 [21]. This representation is expressed with a parameter
t; and from it one can extract the so-called Burau Matrix. The following re-
lationship connects Burau's braid group representation with Alexander's knot
polynomial:
det(Id - Burau Matrix(,6)) = ±tnA,(t) for some n
Burau showed this by connecting the Burau Matrix of )3 with a known
way of calculating A from a presentation of the knot group.
The first firm connection between knot theory on the one hand and braid
theory on the other was made by Alexander in 1923 with a theorem which
showed that any n-link is equivalent to some closed braid. He also gave a simple
algorithm [5] for converting an n-link into a closed braid. The existence of such
an algorithm was already noted by Herman Brunn in 1897 [20]. However,
troubles with this algorithm are twofold. It may cause one and the same p-
link to become a closed braid which, upon cutting can belong to two distinct
braid groups Bn and Bn,, n 0 n'. If the algorithm consistently causes the
n-link to yield a closure of a braid belonging to just one braid group, then
there may be many words representing it.
How does this result affect classification and knot isotopy? Closed braids
have two very interesting properties, which are caught by the so-called Markov
moves.

1. There is a particularly diabolical way of making a knot from an open
braid. If one takes two braids a, Q E Bn and constructs the closure:
(a0a-1 ) then it will equal %3. The closing operation causes a to be
cannibalized by its inverse. This is worded by saying that conjugate
elements in Bn yield equivalent links.
2. Imagine a closed braid on a spar. Adding any number of strands by
means of simple twists cast in any of its outer bights over the spar
does not change the type of link.

The essence of these two properties was captured by A.A. Markov in his
theorem of 1936 which states that closed braids t and ( are equivalent as
links if and only if they can be connected by a finite sequence of elementary
moves, which are precisely those described by the two properties of closed