Topology in Molecular Biology

5 From Tangle Fractions to DNA 77

D(T)

interchange

N(T) =

[0] [ ]

T T =

Fig. 5.10.The [0]–[∞] interchange

N(T)=N(T+ [0])−→N(T+[∞]) =D(T).

We will concentrate on the class ofrational knots and linksarising from
closing the rational tangles. Even though the sum/product of rational tan-
gles is in general not rational, the numerator (denominator) closure of the
sum/product of two rational tangles is still a rational knot. It may happen
that two rational tangles are not isotopic but have isotopic numerators. This
is the basic idea behind the classification of rational knots, see Sect. 5.5.

5.3 Continued Fractions and the Classification of Rational Tangles

of Rational Tangles

In this section we assign a fraction to a rational tangle, and we explore the
analogy between rational tangles and continued fractions. This analogy culmi-
nates in a common canonical form, which is used to deduce the classification
of rational tangles.
We first observe that multiplication of a rational tangleTby 1/[n]may
be obtained as the addition of [n]totheinverse1/T followed by inversion.
Indeed, we have:

Lemma 1.The following tangle equation holds for any rational tangleT.

T∗

1

[n]

=

1

[n]+T^1

.

Thus any rational tangle can be built by a series of the following operations:
Addition of[±1]and Inversion.

Proof.Observe that a 90◦clockwise rotation ofT∗ 1 /[n] produces−[n]− 1 /T.
Hence, from the above (T∗ 1 /[n])r=−[n]− 1 /T, and thus (T∗ 1 /[n])−^1 =
[n]+1/T. So, taking inversions on both sides yields the tangle equation of the
statement.