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A History of Topological Knot Theory 225

a role of utmost importance in topology.
In his efforts to distinguish complexes, Poincare came to introduce torsion
coefficients, and a method for computing Betti numbers of an n-dimensional
complex. These concepts are defined as follows. Given a finitely generated
Abelian group A, it can be written as the direct product of a free Abelian
group F and a family of cyclic groups A/Hi, where each A/Hi is a finite
cyclic group of order hi, and such that h11 h2I ... #A. The rank of the free
Abelian part F and the uniquely defined numbers hi are invariants of the
group A, and completely determine its structure. If A is the homotopy group
in dimension, say d, then the rank of F is the d-th Betti number, and the
hi are the torsion coefficients. They are numerical invariants of isomorphism
classes of finitely generated Abelian groups. The rank of F is used to calculate
the Euler characteristic.
It is interesting to note that Poincare used only methods of continuous
mathematics at the beginning of his series of papers; but by the end he relied
heavily on combinatorial techniques. This was not without impact on the
newly founded schools that formed to take up and develop his ideas. For the
next 30 years researchers concentrated almost exclusively on combinatorial
and algebraic methods.
The belief in the power and aptness of combinatorics ran deep. The
knot problem's solution demanded a formal definition of a knot, which in true
combinatorial spirit became a set of straight arcs making up a closed non self-
intersecting polygon in space. Max Dehn and Poul Heegaard in their article
[29] in Encyklopedie der Mathematischen Wissenschaft in 1907 noted that the
knot problem could be formulated entirely in terms of arithmetic, i.e. combi-
natorics. However this kind of reduction seemed to be of no practical value,
nor did it seem to have any theoretical consequences (e.g. for decidability
of knot equivalences). There are many natural numerical invariants of knots
which may be defined quite easily, such as the already-met number of crossing-
points, the Gordian number, the maximal Euler characteristic and so on; but
difficulties in computing them by solely combinatorial techniques seem to be
inversely proportional to the ease in defining them. There is something general
about this matter. There is for instance, to date, still no known algorithm for
finding the minimal number of crossing points for an arbitrarily given n-link.
In fact there seems to be no hope for finding this number with any tool at all!
On the other hand, a recent attack on the Gordian number has yielded good
bounds for it (1994). A good account of this work, by William Menasco and
Lee Rudolph, can be found in [67].
The first successful algebraic topological invariant attached to a link L
was the fundamental group of the 3-manifold, which is constituted by the link's
complement in 3-space, namely (R3 - L); this invariant is sometimes called
the group of the knot-complement or, simply, the knot group. For an arbitrary