Will a knot tied in a hanging, frictionless rope slip out under the force of gravity? I am overall just curious about what keeps knots where they are in a rope. Another related question you might be able to answer is: What happens if you tie a bowline on the bight in a frictionless rope and pulled on it, would it then slide together? I doubt it, but I'm not quite sure.
For clarity, here's the bowline on a bight knot: https://www.netknots.com/rope_knots/bowline-on-a-bight/
 A: Let's consider the following (simplified) free body diagram with no friction force:

Since a rope is a single body, the force $\vec F$ you use for pulling the rope to tighten it is offset by the tension $\vec T$. If we suppose a frictionless rope and the situation in which you are not acting over the rope, it turns out that the entire $\vec F$ is done by the gravity force and, in some moment, the rope reaches an equilibrium, that is, the vector sum of all the forces acting over it is zero. What happened? It is true that there was not friction force (apparently), but if you look at the yellow region, it is necessarily performing a force upward since the green region is applying a force downward (its weight).
Finally, the rope does not slide since the pink region is applying a force against the section of rope which contains the green region (due to $\vec T$), since the green region is doing the same against the region of rope which contains the pink region (due to $\vec F$), and since the yellow region is supporting the weight of the green region.
More: I would say that it doesn't make sense to consider a frictionless rope, at least in this situation, because as I have pointed out, we have three contact points... but what does that "contact points" really mean? Friction force is an electromagnetic phenomenon in which the particles repel to each other, and macroscopically we can only see a kind of resistance between two bodies when they contact in some way (i.e. when rubbing to each other).
It's like to ask: if we drop an apple from the air and we consider no friction forces between it and the ground, will the apple reach the center of the Earth? Well, the apple has atoms, the Earth has atoms, so the electromagnetic force must exist between them, otherwise the ground would not "stop" the apple when they were at the same level, and the apple would keep on falling until it reached the center of the Earth due to the gravity force it experiments (the unique one).
