Topology in Molecular Biology

90 L.H. Kauffman and S. Lambropoulou

5.4.2 The Fraction Through Colouring

We conclude this section by giving an alternate definition of the fraction
that uses the concept of colouring of knots and tangles. We colour the arcs
of the knot/tangle with integers, using the basic colouring rule that if two
undercrossing arcs colouredαandγmeet at an overcrossing arc colouredβ,
then α+γ=2β.We often think of one of the undercrossing arc colours as
determined by the other two colours. Then one writesγ=2β−α.
It is easy to verify that this colouring method is invariant under the Rei-
demeister moves in the following sense: Given a choice of colouring for the
tangle/knot, there is a way to re-colour it each time a Reidemeister move is
performed, so that no change occurs to the colours on the external strands of
the tangle (so that we still have a valid colouring). This means that a colour-
ing potentially contains topological information about a knot or a tangle. In
colouring a knot (and also many non-rational tangles) it is usually necessary
to restrict the colours to the set of integers moduloNfor some modulusN.
For example, in Fig. 5.16 it is clear that the colour setZ/ 3 Z={ 0 , 1 , 2 }is
forced for colouring a trefoil knot. When there exists a colouring of a tangle
by integers, so that it is not necessary to reduce the colours over some modulus
we shall say that the tangle isintegrally colourable.
It turns out thatevery rational tangle is integrally colourable:To see this
choose two “colours” for the initial strands (e.g. the colours 0 and 1) and
colour the rational tangle as you create it by successive twisting. We call
the colours on the initial strands thestarting colours. (see Fig. 5.17 for an
example). It is important that we start colouring from the initial strands,
because then the colouring propagates automatically and uniquely. If one
starts from somewhere else, one might get into an edge with an undetermined
colour. The resulting coloured tangle now has colours assigned to its external
strands at the northwest, northeast, southwest and southeast positions. Let
NW(T),NE(T),SW(T)andSE(T) denote these respective colours of the
coloured tangleTand define thecolour matrix ofT,M(T), by the equation

M(T)=

[

NW(T)NE(T)

SW(T) SE(T)

]

.

0

1 234

0

1 23

0 = 3

α

β

2β − α

4

1 = 4

Fig. 5.16.The colouring rule, integral and modular colouring