Topology in Molecular Biology

5 From Tangle Fractions to DNA 91

or

0 0

0

0

11

1

1

1

-1^1

1

0

0

0

2

0

1

23 4

3

-3

-6^11

18

T = [2] + 1/([2] + 1/[3])

F(T) = 17/7 = f(T)

Fig. 5.17.Colouring rational tangles

Definition 4 To a rational tangleTwith colour matrixM(T)=

[

ab

cd

]

we

associate the number

f(T):=

b−a

b−d

∈Q∪∞.

It turns out that the entriesa, b, c, dof a colour matrix of a rational tangle
satisfy the “diagonal sum rule”: a+d=b+c.

Proposition 1 The numberf(T)is a topological invariant associated with
the tangleT. In fact,f(T)has the following properties:

1 .f(T+[±1]) =f(T)± 1 ,

2 .f(−T^1 )=−f(^1 T),

3 .f(−T)=−f(T),

4 .f(T^1 )=f(^1 T),

5 .f(T)=F(T).

Thus the colouring fraction is identical to the arithmetical fraction defined
earlier.

It is easy to see thatf([0]) =^01 ,f([∞]) =^10 ,f([±1]) =± 1 .Hence State-
ment 5 follows by induction. For proofs of all statements above as well as for
a more general set-up we refer the reader to our study [13]. This definition is
quite elementary, but applies only to rational tangles and tangles generated
from them by the algebraic operations of “+” and “∗”.
In Fig. 5.17 we have illustrated a colouring over the integers for the tangle
[[2],[2],[3]] such that every edge is labelled by a different integer. This is al-
ways the case for an alternating rational tangle diagramT.For the numerator