Linking Number

Carl Fredrich Gauss was an alien. Or at least, sometimes I feel that way when I look at his writings.
While reading some math online inspired by a conversation with my friend Nick, I recently came across Gauss' first mention of the linking number of two smooth closed curves in $\mathbb{R}^3$:

I vaguely remembered seeing this crazy - looking formula at some point in the past, but never really thought about where exactly it comes from. I had a bunch of knot theorist friends in graduate school but am not one myself, so figured like many things, the best way to try and really understand something like this is to figure out how to derive, or figure out how to fit it into a more general (hopefully less coordinate-heavy/scary-looking) story.
Below are my notes from working this out: I’ll hopefully turn these into a more coherent story when I have some time!

A warning to potential readers: this is almost certainly not the fastest / most efficient way to get to the heart of the matter, but rather is just the one that made most sense to me. Also: if the pdf is not loading below, you can access it directly here.