Now I’m wondering if there is a way to use the intersection points of the curves as bezier curve paths, which are the curves that give a 3D knot the basic shape.

The edges of a mesh like that form a graph.
It has nodes each with multiple edges connected.

A knot is an embedding of a circle - it doesn’t have any branching points.

You can use a graph as a starting point for building a knot or link (like knots but with multiple components), in various ways, but it isn’t clear to me what you are after.

You can certainly select any number of points in space(from a mesh or otherwise) and connect them into a single smooth closed curve. This curve might be knotted and it might not.

(By the way a torus knot means something specific in knot theory. I’m guessing you’re not looking for torus knots specifically)

Is the question about how generally to make nice looking knot curves in space, or something more specific?

Not specifically. It could be interesting to also work with knot theory, but there is no need. I just wrote “torus knot” because of the typical visual appearance on images. I just like the style.

Yesss, “nice looking knot curves in space” is exactly what I am looking for. And nothing specific.