Linking number and Möbius energy
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We gave several defintions of the linking number Lk(A,B) between a pair of
disjoint oriented A and B loops in R^3:
• half the number of (signed) crossings of A with B in a generic projection
[the ± sign jibes with the right/left handedness of the oriented crossing's
"screw thread"]
• the number of (signed) crosings of A over B (or of B over A)
• the degree (signed number of generic preimages) of the Gauss map
g(s,t) := A(t)-B(s)/|A(t)-B(s)|
from S^1 x S^1 to the "celestial" 2-sphere S^2
• the (signed) area of the Gauss map image, divided by the area of S^2
[A useful slogan, when the link lies near the plane of projection: each
crossing contributes to Gauss image a hemisphere whose pole is normal to
the plane of projection, i.e. contributes ±2π to signed Gauss image area.]
• the Gauss integral formula for linking number using triple product:
Lk(A,B) = (1/4π)∫∫g_t¢g_s•g dt ds
= (1/4π)∫∫A'(t)¢B'(s)•A(t)-B(s)/|A(t)-B(s)|^3 dt ds
(as before, I'm typing ¢ for cross product, but please use the usual cross).
We also defined the Möbius "cross" energy for a pair of loops in R^3 (or in
S^3 in R^4) with a double integral similar to the Gauss linking formula:
E(A,B) := ∫∫|A'(t)||B'(s)|/|A(t)-B(s)|^2 dt ds
This energy is clearly scale-invariant, but also invariant under all Möbius
transformations of space. Just as euclidean transformations are generated
by reflections, Möbius transformation are generated by inversions, which
can be thought of as "reflections in round spheres" with the simplest being
inversion in the unit sphere at the origin, which takes an arbitrary point X
in space to the "inverted" point i(X):=X/|X|^2; note that i interchanges
"the point at ∞" with 0; also note that stereographic projection is the
restriction of such an inversion (the "reflection sphere" is centered at the
north pole and meets the equator of the domain sphere).
Project 4a:
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Explain why these definitions of linking numner are equivalent, explore the
basic properties (such as Lk(A,B) = Lk(B,A) = -Lk(A,B*), where B* is the
reverse-oriented loop), and compute Lk(A,B) directly for some links you can
parametrize explicitly (beyond the Hopf and Whitehead links we drew in class).
Project 4b:
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Show that the integrand of the Möbius energy is not only scale-invariant, but
also invariant under all Möbius transformations - it's enough to show it's
invariant under inversion i(X)=X/|X|^2 in the unit sphere.
Also, fill in the details of our argument in class that E(A,B)≥ 4π|Lk(A,B)|
for a general link with 2 components A and B, and explicitly compute E for
the "geometric" Hopf link we drew in class:
A = straight line (i.e. a circle through ∞), and
B = unit circle in perpendicular plane centered on this line.
This link happens to be the stereographic image of the Hopf link in S^3
formed by two mutually perpendicular 2-planes (like the complex lines
{z=0} and {w=0} in R^4=C^2) and in this picture |A(t)-B(s)|=√2 for all
s and t, so the integrand is constant! Because stereographic projection
from S^3 to R^3 extends to a Möbius transformation of R^4 to R^4, the
Möbius energy for either picture is the same!
Project 4c:
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The most general "geometric" Hopf link in S^3 has k components A_1,...,A_k,
where each A_i is the intersection of a complex L_i line in C^2=R^4 with S^3.
Each line has the form L_i={za_i+wb_i=0} whose slope a_i/b_i is a point on
C \union ∞ = R^2 \union ∞ = S^2; this defines the Hopf projection S^3 to S^2
(there are nicer ways to do this using quaternions, sketched in class...).
This k-component Hopf link projects to a k-configuration U=(u_1,...,u_k)
on S^2. For the example in 4b, we get u_1 at the north pole and u_2 at
the south pole (since L_1={z=0} and L_2={w=0} implies a_1/b_1 = 1/0 = ∞
and a_2/b_2 = 0/1 = 0).
Show that the Möbius (cross) energy E(A_1,...,A_k)=\sum_{i