Topology in Molecular Biology

184 M. Monastyrsky

Proposition 7.The integrals
∫

B 1

u ̃ 123 =−

∫

B 2

u ̃ 123 =

∫

S^3

̃v′ 123 =k 2 (l) (10.11)

have integer values and do not depend on the choice ofu 12 ,u 23 ,v 12 ,v 23 in the
corresponding cohomology class.

Let us consider sublinks of three curveslijk=(li,lj,lk)for1≤i<j≤nof a
linkl=(l 1 ,...,ln) withn≥3. We introduce the second linking number forl

̄k 2 (l) = max

1 ≤i<j<k≤n

|k 2 (lijk)|.

Example 1.For Borromean ringsk 2 (l)=1.Iflis homotopically unlinked, e.g.,
Whitehead link, then the number ̄k 2 (l) is zero. But there exist the links with
̄k(l) = 0, but homotopically unlinked. To characterize such links it is possible

to define a linking number of order three ([2, 5]).

10.3.2 Formula Cˇalugˇareanu and Supercoiled DNA

Letγbe a closed smooth curve inR^3 andva normal vector field onγ.vis a
one-parameter family of oriented line segments, one end of each which lies on
γ, so thatvforms a ribbon. We choose the length ofγto be so small that the
line segment meets the curveγonly at its initial point, so that the ribbon is
embedded. The curve of the endpoints ofvγvinherits the orientation of the
curveγ.Letk(γ,γv) be the Gauss linking number ofγandγv. We define also
the total twist ofv
tw=

1

2 π

∫

v⊥dv

in a standard way, where the vectorv⊥is in the right-handed frame (ˆt, v, v⊥),ˆt
is the unit tangent vector to the curveγ. The twist of the curve is a continuous
quantity, the linking number is integer, thereforek=tw. It is remarkable that
another quantity can be introduced, viz.,k−tw=Wr(the so-called writhing
number) such that
k=tw+Wr.

Wronly depends onγand its definition is based on the following. We con-
struct the Gauss map forγ×γ, i.e., we define

φ:γ×γ→S^2 ,φ(s, y)=

y−x

|y−x|

for ordered pairs (x, y). Let dsbe the element of area onS^2. Then the form
φ∗(ds) is induced byφ.

Definition 4.The writhing number is the integral

Wr=

1

4 π

∫

γ

∫

γ

φ∗(ds). (10.12)

Wris a continuous quantity.