A History of Mathematics From Mesopotamia to Modernity

Modernity and itsAnxieties 227

Type I Reidemeister move

Type II Reidemeister move

Type III Reidemeister move

Fig. 8The three Reidemeister moves. Type I flips a loop over; type II pulls one looped strand over another; type III takes a strand
through a crossing.

equivalence’. Given this definition, Reidemeister was able to show that two knotprojectionswere
equivalent if and only if you could go from one to the other by a sequence of ‘Reidemeister moves’,
as shown in Fig. 8.
On the face of it this is a rather unambitious result; and indeed it was a rather small basis for
the construction of more and easier invariants by Reidemeister and Alexander. However, again it
illustrates a major change in the way mathematicians treat their subject-matter. For Tait, a knot
is obvious; it is represented by a picture, and we know what it means to say that two knots are
the same. For Reidemeister, we have to construct meticulous (and finitistic!) definitions both of
the object ‘knot’ and of the relation ‘the same’. The payoff is a relation between diagrams which
guarantees sameness. This too may not be easily checkable; however, if we construct a function of
a knot diagram (as several authors were to do in the 1980s), we can show that we have a ‘knot
invariant’ by showing that the function does not change under the three Reidemeister moves. The
study of knots has become a kind of algebra.

Exercise 3.
(a) Show that the two knots shown (Fig. 9) are equivalent directly.
(b) Show that they are equivalent using Reidemeister moves. How many have you used?