Page 235 - J. C. Turner "History and Science of Knots"

226 History and Science of Knots

link L, and with reference to a basepoint p in L's complement, the knot group
is denoted by 7rl(R3 - L, p). This group is one in which the elements are
homotopy classes of (unknotted) loops which traverse the complement space
of a knotted structure, starting from and terminating at the basepoint p. The
binary operation for the group is the composition of two loops, carried out
by concatenating them at p. Since this composition is non-commutative, the
fundamental group (knot group) is non-Abelian.
The fundamental group expresses in algebraic language some of the topol-
ogy of the knot-complement, which makes it possible to compare different
knots by comparing their algebraic descriptions. A knot's complement, which
is three-dimensional, carries a richer topological structure than the knot it-
self, which is one-dimensional. The topological structure of the complement
necessarily contains certain information about the knot. In 1908 Tietze con-
jectured that it contained all such information; an idea that did not become an
established fact for 1-links until 1988 [40]. The uncertainties surrounding this
conjecture did not prevent this avenue being pursued vigorously; presentations
of certain knot groups appeared fairly soon in the literature. General methods
for writing down a presentation of the knot group from a knot projection were
introduced by Wirtinger, who did not publish them; but he got credit for the
idea anyway [65]. Max Dehn, in 1910, also published methods for presenting
knot groups.

7. Max Dehn's Work in Knot Theory

It was thought that by considering the knot groups one might be able to classify
knotted structures. The initial notions on groups had arisen from 19th-century
algebra, analysis and geometry. By the time that Max Dehn began his work
on knots, early in the 20th century, group theory had proceeded so far that
it was no longer necessary to describe groups by means of their cumbersome
Cayley (composition) tables. At the beginning of the 1880's von Dyck had
shown how every group is the homomorphic image of a free group, and how
one could present such a group by giving so-called generators and defining
relations . Armed with these tools Dehn attacked the knot group.
In his 1910 paper Ober die Topologie des dreidimensionalen Raumes [27],
Dehn discussed a method for extracting a description of the knot group, the
so-called Dehn presentation. He did so by the following algorithm:

1. List and denote all bounded regions of a knot-diagram by C1i ... , Cn.
These are to be considered the group generators.

2. Every over-crossing yields a relation R,, 1 < i < n by noting down a
relation containing a sequence which is the product of generators as