Topology in Molecular Biology

10 Hopf Fibration and Its Applications 179

(See the proof in [1] or [2].)
We now investigate different properties of Hopf fibrations.

10.1.2 Linking Numbers

Letfbe a smooth mapS^3 →S^2 ,y 0 ,y 1 two regular points inS^2 andM 0 and
M 1 the preimages ofY 0 andy 1 equal tof−^1 (y 0 ),f−^1 (y 1 ), respectively. We
defineH(f)={M 0 ,M 1 }to be the linking number for the inverse images. It
follows from the definition of regularity thatf−^1 (yI)∼S^1.

Definition 1.The linking number (or coefficient) of two disjoint curves
γi(t)i=1, 2 lying in the Euclidean spaceR^3 (γi(t)=ri(t)) (0≥t≥ 2 π, ris
the radius-vector of a point inR^3 ) is the number

{γ 1 ,γ 2 }=

1

4 π

∫

γ 1

∫

γ 2

〈[dr 1 ,dr 2 ],r 1 −r 2 〉

|r 1 −r 2 |^3

. (10.5)

Here〈〉and [ ] denote scalar and vector product, respectively. It is possible to
determine linking number coefficient in terms of homology. It will be followed
from the equivalent definition the integer-valued of{γi,γj}in (10.5).
Let us remind one important definition. We require several facts from the
intersection theory, which we determine in more general setting.

10.1.3 Intersection Number

LetPrandQsbe two closed submanifolds ofMnof dimensionsr ands,
respectively. By the classical theorem of Whitney we can always regardMas
an Euclidean space of a sufficiently large dimension.
Pris said to intersect transversally toQs(or to be in general position) if
at any pointx∈Pr∩Qs, the tangent spacesTxPrandTxQsgenerate the
tangent space ofMn.
In particular, it follows that in general position the intersectionPr∩Q^3
is a smooth (r+s−n)-dimensional submanifold.
Example 1. A straight lineP^1 and the planeQ^2 inM^3 intersect transver-
sally if P^1 does not meet Q^2 at the zero angle, i.e., is not in Q^2 .If
dimPr+dimQs=dimMnthen in general position the manifoldsPrandQs
at one or more points. IfMn,Pr,andQsare oriented, then each intersection
pointxiis given a sign by the following rule: letτjrbe the oriented tangent
frame toPrat the pointxjandτjsoriented frame toQsatxj. The pointxj
is assigned a plus sign if the union frame (τjr,τjsis orientating forMnatxj.
Otherwise, a minus sign is given. The sign is denoted by sgnxj(P◦Q).

Definition 2.The intersection number of two manifolds,PrandQs, is the
integer

Ind(P◦Q)=

∑m

j=1

sgnxj(P◦Q), (10.6)

where m is the number of intersection points.