Page 242 - J. C. Turner "History and Science of Knots"

A History of Topological Knot Theory 233

obtained by projection of the knot into a plane. He discovered that there were
three fundamental changes (and their inverses); they are shown in diagram
form below (Fig. 16). The left-hand sketch shows how a loop can be untwisted
(removing one crossing); the centre one shows the pulling apart of two flaps
(removing two crossings); and the right-hand one shows a portion of string
passing over a crossing point (leaving the number of crossings unchanged).
These three types of change, together with their inverse changes, are known
as the Reidemeister moves. It should be evident that none of these changes
relates to a change in the topological nature of the underlying knot.

Fig. 16. The Reidemeister moves
The importance of the Reidemeister moves in combinatorial knot theory
is embodied in the following key theorem:

If two knots (or links) are topologically equivalent, their diagrams
can be transformed one to the other by some (finite) sequence of
Reidemeister moves.

It should be noted that in any given case, there are many (indeed an
infinite number of) different sequences of the three Reidemeister moves and
their inverses which effect a transformation from one diagram to the other.
Reidemeister published the first book*on knot theory, in German, in 1932:
an English edition of this book was published in 1983 [74].
Midway during the 20th century the history of knot theory, like much
else, was temporarily disrupted by a seemingly global desire to practice politics
*J. B. Listing wrote the first book [621 on topology in 1847; it was dedicated primarily to
knot theory. Bernhard Riemann was a student of Listing, and he learned about knots from
Listing's book.