A cow hitch is not a knot, in that it can be easily transformed into an 'unknot'.
But it's clear to me that the utility of a cow hitch is all in it's topology. Two infinite parallel lines joined by a cow hitch can't be unhitched without cutting the string.
I imagine that one way to pose the question would be: What are all of the knots that exist on a genus 2 solid torus? As the cow hitch is no longer trivial knot when connecting the two holes on a solid torus of genus 2.
In sum my question is: What should I read about in knot theory to understand various ways to hitch two infinite non intersecting lines together
Best Answer
As was pointed out in the comments, you can also consider this as a 3-component link (your knot component, and then two more components: one passing through each of the two holes in the surface). The fact that this link is not splittable can be shown, for example, by the Alexander polynomial (since the Alexander polynomial of a splittable link is always 0).
You might also be interested in virtual knots - these can be defined as stable equivalence classes of knots on thickened surfaces, so that your cow hitch also represents some virtual knot (presumably a non-trivial one).