Topology in Molecular Biology

6 Writhe Versus the Number of Crossings 117

crossings gives rise to the unknot. One cannot nullify the last crossing without

disconnecting the link. Sowx=a−1=n−1andwy= 1. Using (6.1) we get:

PWr=

10

7

(n−1) +

4

7

=

10

7

n−

6

7

. (6.2)

aEven

Figure 6.3c shows that the closure of such a tangle gives rise to a two-

component link. There are therefore two possible orientations for the second

component.

(a) Family of links 4^21 +−, 621 ++, 821 + +, etc. (The symbols + and−refer

to orientation. The convention used can be found in [13].) Figure 6.3d

shows the nullification process applied to those links. Here again, we get

wx=a−1=n−1andwy= 1. Therefore

PWr=

10

7

(n−1) +

4

7

=

10

7

n−

6

7

. (6.3)

(b) Family of links 4^21 ++, 621 +−, 821 +−, etc. Figure 6.3c shows that the
acrossings are now of negative sign and that only one crossing may be
nullified. We already reach the unknot, and any further nullification would
create a disconnected component. Thuswx=−1andwy=−(a−1) =
−(n−1). That is to say, we have nullified one negative crossing and there
remainsn−1 negative crossings. Using (6.1) again, we get:

PWr=−

10

7

−

4

7

(n−1)

=−

4

7

n−

6

7

. (6.4)

6.3.2 Tangles with Two Rows, Denoted by (a)(b),aandbPositive

Integers

Now,n=a+b. Figure 6.3a shows an example of such a tangle, witha=3

andb=2.