Question

I found something interesting about a figure 8 knot and its relation to sine and cosine, any advice or ideas on where to go from here? One sec while I draw it out.

Answer 1

So if you just tie a figure 8 knot, you can continuously deform it like this. So if we start at the top left we can deform it once by flipping stuff around one way to get the next one.

But let's pretend each knot is a function, so we can call a knot f(x). Let's say f(-x) is what happens when you reverse the string direction by 180 degrees

Now we'll also say that reversing all the crossings takes us from f(x) to -f(x). Also, notice that for this overhand knot f(x)=f(-x).

Ok one more thing to show you what I'm talking about...