Advanced book on Mathematics Olympiad

20 1 Methods of Proof

A deep theorem of Reidemeister states that two diagrams represent the same knot
if they can be transformed into one another by the three types of moves described in
Figure 8.

(I) (II)

(III)

Figure 8

The simplest knot invariant was introduced by the same Reidemeister, and is the
property of a knot diagram to be 3-colorable. This means that you can color each strand
in the knot diagram by a residue class modulo 3 such that

(i) at least two distinct residue classes modulo 3 are used, and
(ii) at each crossing,a+c≡ 2 b(mod 3), wherebis the color of the arc that crosses
over, andaandcare the colors of the other two arcs (corresponding to the strand
that crosses under).
It is rather easy to prove, by examining the local picture, that this property is invariant
under Reidemeister moves. Hence this is an invariant of knots, not just of knot diagrams.
The trefoil knot is 3-colorable, as demonstrated in Figure 9. On the other hand,
the unknotted circle is not 3-colorable, because its simplest diagram, the one with no
crossings, cannot be 3-colored. Hence the trefoil knot is knotted.

0

2

1

Figure 9

This 3-colorability is, however, not a complete invariant. We now give an example
of a complete invariant from geometry. In the early nineteenth century, F. Bolyai and a