# How does knot theory deal with a cow hitch?

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

• Yes, you could consider it as a knot in a solid torus of genus $2$, depending on what you do with the ends which I can't quite figure out from the pictures on Wikipedia. You can also consider the solid circle(s?) together with the cow hitch as a link with two (three?) components: en.wikipedia.org/wiki/Link_(knot_theory) Commented Aug 28, 2022 at 7:30
• @QiaochuYuan, as a link it can actually be undone. That's why I specified that the circles need to be solid. Commented Aug 28, 2022 at 7:41
• Ah, I see. Then I don't have a better interpretation than a knot in a solid torus, but I also don't know where you'd go to find anything about such knots. Commented Aug 28, 2022 at 7:45
• Really? If the rope is long enough (compared to the sizes of the circles), then the cow hitch can be undone, right? Commented Aug 28, 2022 at 7:54
• @BenjaminWang I edited the question to include your stipulation. Commented Aug 28, 2022 at 8:04

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).