226 A History ofMathematics
Around 1910, Dehn and Wirtinger were aware of the tables of knots (or knot-projections)
compiled by Tait, and could see that beneath them a question of topology, treatable by Poincaré’s
new methods, might lie. The problem was that the knot K was just a circle, the ambient space just
R^3. The answer was to consider the ‘difference’,R^3 −K. Not a ‘manifold’ in Poincaré’s sense, since
it was infinite, this still seemed a good subject for treatment. It is attractive in philosophical terms
to note that the first step forward (rather like Dedekind’s?) replaced studying the knot by studying
the hole which was left when you removed it.
One of Poincaré’s most interesting invariants was agroup, which we now call the ‘fundamental
group’ ofX,π 1 (X); and he had given a means of computing it from a cell decomposition. Although
the ‘cells’ inR^3 −K were not obvious, Dehn and Wirtinger did arrive at a description of generators
and relations for the groupπ 1 (R^3 −K).^11 An excellent first stage, this ran up against serious
problems relating to how little was known about such presentations. When did two define the same
group? Was the problem even (in intuitionist, or Gödelian terms) decidable? (It is not.) Information
can be gathered about the group when you are lucky, but how can you enforce luck?
The next major advances were due to Alexander and Reidemeister in the 1920s. There may be
various priority questions to disentangle here, on which Epple has commented (2004), but they
need not concern us here. The first point is that a new definition of a knot was found useful.
‘Simplexes’ (triangles, tetrahedra, etc.) were seen, correctly, to be a more precise way of finding
your way around than Poincaré’s more general ‘cells’; and so a knot K was defined to be a closed
polygonin three dimensions. An ‘elementary equivalence’ was defined to be one of the type shown
in Fig. 7 where you replaced the side AB of the triangle by the sides AC, CB, or vice versa—provided
that K did not meet the interior of the triangle. And, finally, K and K′were equivalent if you could
get from one to the other by a sequence of elementary equivalences.
This was a substantial change. Was it proved equivalent to the previous definition? I am not
sure, and in a way it is not so important as the new language. While the new knots look pretty
much like the old, and in actual drawing topologists often smooth out the corners for reasons of
aesthetics or laziness, there is still a greater precision which is involved specifically in the ‘elementaryABC1Fig. 7Elementary equivalence. The shaded triangle ABC does not meet the other strands of the knot, so AB can be replaced by the
two edges AC, CB.- That is,π 1 is the set of all expressions in (say)x 1 ,x− 11 ,...,xk,x−k^1 (generators), subject only to the rules which follow from
requiring certain expressionsR 1 ,...,Rlin thexs (relations) to equal 1. See a book on group theory.