Topology in Molecular Biology

10 Hopf Fibration and Its Applications 185

Proposition 8 (Formula Cˇalugˇareanu).

k(γ,γv)=tw+Wr. (10.13)

It is useful to compare the formula (10.5) for linking number and the
formula (10.12) for writhing. The formal difference is that in the first case we
integrate untegrand (10.5) over the spaceγ×γvand in the second one over
the spaceγ×γ. So the writhing number characterizes the single curveγ.

Remark 1.Wris a very important quantity since in experiments with DNA
moleculeWrcan be measured directly, whilekandtwcannot be [22]. Different
applications of formula (10.13) will be discussed in several chapters of our
book. These applications and the modern proof of formula Cˇalugˇareanu can
be found in the book [6]

Remark 2.Sometimes formula (10.13) is called formula Cˇalugˇareanu-White,
or simply White formula. White proved in 1968 the multidimensional gener-
alization of formula (10.13). In the article [23] he very clearly explained the
relation between his result and Cˇalugˇareanu’s [7]. But lately, in modern lit-
erature, the name of Cˇalugˇareanu gradually disappeared. (In this connection
see the paper [8]).

Remark 3.The Cˇalugˇareanu formula implies that a ribbon swept byvforms
a simple knot, i.e., the linking numberk(γ,γv) is the only topological invari-
ant. It would be interesting to modify formula Cˇalugˇareanu in the case when
the first linking number is zero and there exists next nontrivial second-order
coefficient.

The question arises as to whether there are similar formulas for a linking
ensemblel=(l 1 ,l 2 ,...,ln).

10.3.3 Hopf Fibration and Membranes

One more example where Hopf fibration appears in physics and biology is the
problem of phase transition in liquid membranes. We remind the mathematical
essence of this problem. Applications in biological and physical systems can
be found in [9, 10]. LetM^2 be a compact surface embedded inR^3 andFthe
functional

F=

∫

M^2

H^2 dA, (10.14)

whereHis the mean curvature.

Problem 1.To determine all compact surfaces of fixed genusgthat minimize
the functional (10.14)
δF= 0 (10.15)