I apologize if this is an easy thing to do and I'm just failing at finding the right search terms. Everything I've searched for has just resulted in showing me how to do my first example.

If I have two paths and use the division tool I can neatly cut one path by another, having duplicates of the path cut by eachother can let me create a celtic knot like pattern.
 

Is there a simple way, without a lot of precutting, to achieve the same effect with a single path that intersects itself?

I realize I could break it apart node by node so I end up with loads of overlapping lines, or go through and manually add nodes at every intersection, and with my example that would work, but with a more complicated path that would take a long time.

Is there any way to achieve the path division result, or even the path cut result with a single path that overlaps itself?

Here's a lateral-thinking workaround.

On reflection I realise I haven't answered your question. I presented a neat solution to a totally different problem.

I suggest playing with [Path Effect > Knot] to create a gap at each path intersection. At each intersection you can select the upper and lower segment (i.e. which segment to break). Unfortunately this won't give you a perfect over/under effect because a square cut is applied to the lower line segment.

I can't think of a simple technique so here's a workaround. 

I like the third strategy, that would definitely work. 

I think I came up with something this morning, though, that seems to work as with a little cleanup:

starting with the Knot path effect, then adding "Pattern Along Path" to two short lines that have been combined into one object, then duplicating the original line to act as a fill (all my attempts to fill it in the pattern or in the final path ended up giving unwanted results)

It does still have a little cleanup at sharper corners and lower angled intersections, I think between your answers and this I might just have what I need. It's unfortunate that knot doesn't have some way to bevel the parts where it cuts for the gap, it would be absolutely perfect.

Thank you for your help!!

After many experiments, I have an improved solution.

https://inkscape.org/forums/tutorials/dynamic-knots/