182 M. Monastyrsky
10.3.1 Generalized Linking Number
Consider the following problem: What kind of topological invariants would
make it possible whether one can decouple (using motions inR^3 ) a system of
linked closed curves (loops). A classical invariant of this type is the Gauss-
linking coefficient (10.5) of two loops. But knowing this coefficient is not
enough to solve the problem of decoupling. Well-known examples such as the
Whitehead link and Borromean rings (Fig. 10.2a, b) show that the condition
kG(l 1 ,l 2 ) = 0 gives only a necessary condition for decoupling the two curves.
Such invariants are the high-order linking coefficients, which generalize the
Gauss coefficients and as constructed in [5]. We show what the high-order
linking coefficients look like in the simplest case of curvesl=(l 1 ,l 2 ,l 3 )em-
bedded inS^3. Let us begin with the coefficientk(l 1 ,l 2 ) for two curvesl 1 andl 2.
As follows from section I the coefficientkcan by defined by the two equivalent
definitions:
- As the intersection number Ind(z,l 2 ) of two-dimensional circlez(zis a
film spanned on the curvel 1 ) withl 2 or - In the integral form (10.5)
The linking coefficientk(l 1 ,l 2 ) defined in (10.5) can be calculated via dif-
ferential formsu 1 ,u 2 defined on the curvesl 1 andl 2. Formsuiare defined
by means of Alexander duality, which we determine in the special case one-
dimensional sets embedded inS^3.
Proposition 6 (Alexander duality).LetKbe a one-dimensional compact
set,K⊂S^3. There exists the isomorfphismf
H 1 (S^3 \K)=H^1 (K).
Let us apply the Alexander duality in the casek=l=(l 1 ,l 2 ). The dif-
ferential Alexander-dual 1-formsuiis defined in the complement toli, closed
and characterized by
∫
cui=k(c, li) for any closed curve from the complement
ofliinS^3. The cohomoly class ofuiis determined uniquely.
Now letBi(i=1,2) be the boundary of tubular neighborhood oflinot
meeting another curve. Then
Fig. 10.2.(a) Whitehead link, (b) Borromean rings