Topology in Molecular Biology

5 From Tangle Fractions to DNA 75

Tr T

r

=

= -1/T , - = 1/T

T S

T

S

, , -T -T

T+S

T*S

Ti

T ~

i

=

Fig. 5.6.Addition, product and inversion of two-tangles

180

hflip

R

R

R

vflip

R

o

180 o

Fig. 5.7.The horizontal and the vertical flips

two-tangleTby 1/TorT−^1 ,and hence the rotation of the tangleTcan be
denoted by− 1 /T=−T−^1.
We now describe another operation applied on two-tangles, which turns
out to be an isotopy on rational tangles. We state thatRhf lipis thehorizontal
flipof the tangleRifRhf lipis obtained fromRby a 180◦rotation around a
horizontal axis on the plane ofR.Moreover,Rvflipis thevertical flipof the
two-tangleRifRvflipis obtained fromRby a 180◦rotation around a vertical
axis on the plane ofR(see Fig. 5.7 for illustrations). Note that a flip switches
the endpoints of the tangle and, in general, a flipped tangle is not isotopic
to the original one.It is a property of rational tangles thatT∼Thf lipand
T∼Tvflipfor any rational tangleT.This is obvious for the tangles [n]and
1 /[n]. The general proof crucially uses flypes, see [13].
The above isotopies composed consecutively yieldT∼(T−^1 )−^1 =(Tr)r
for any rational tangleT.This shows that inversion (rotation) is an operation
of order 2 for rational tangles, so we can rotate the mirror image ofTby 90◦
either counter-clockwise or clockwise to obtainT−^1.
Note that the twists generating the rational tangles could take place be-
tween the right, left, top or bottom endpoints of a previously created rational