This supplants an earlier more tedious derivation of 4π.
I later learn that the following is equivalent to Gauß’s 1833 ‘degree of map’ proof to which I was probably exposed as a student.
We want the double integral of
[di dj r]/|r|3
where di is an infinitesimal vector element of a loop and
dj is likewise for a different loop linked to the first.
The difference between the locations of the two elements is the vector r.
Pick a direction from which to view the linked loops and consider 2D projections such as this:
For such a projection view each loop in the plane as a one way road where intersections are underpasses and overpasses.
Choose the black road and traverse it once noting each time you pass over the red road.
If the traffic on the red road is moving to your right increment a counter, if to the left, decrement it.
At the end the net count will be 0 if the loops are unlinked, and ±1 if they are simply linked.
If the viewing direction is changed by a small amount, the net count is unchanged — indeed it is the same for all directions but a set of measure zero.
This is a topological argument so far.
We now make it quantitative.
We claim that the net count times 4π is the double integral we introduced above.
Consider a point p on the red loop and a point q on the black loop.
The ray starting at p and going thru q will pierce a large sphere centered on the loops and a point s.
If we call our space a vector space, and use the sphere about the origin with radius 1, and take p and q as vectors, then the point on the sphere is (q−p)/|q−p|.
Each point on the sphere produces a projection as above.
If we choose a small interval on each loop and points p and q vary on those respective intervals, then the points s will compose a small parallelogram on the sphere whose (oriented) area is
[di dj r]/|r|3 where r = |q−p|.
The content of the sphere is 4π and each part of the sphere contributes one time because the net crossovers seen from each point is 1.
Such an analysis of the reflexive trefoil fails since some small changes in perspective change the self crossing numbers by an odd amount.
This sheds further doubt on the existence of constants for trefoil induction.
See this about higher dimensional linking.