An open trefoil is a trefoil tied in an infinitely long line. An open trefoil that is at rest in (flat) space is a surface in space-time. Now, other observers in space-time have other time slices/cuts. These slices are unbounded but need not be flat surface; they can be curved, because observers can be accelerating. 

The question: Which types of knots may such observers see, if the observer at rest sees an open trefoil? Or: which types of knots can appear when the space-time surface of the open trefoil times R is cut/sliced?

Probably, this is an infinite family of knots; but is there a simple way to find out the simplest cases (that is, those with the smallest crossing numbers)?

Thank you a lot for any help.

A clarification added later:

- Yes, the question should indeed be generalized to any non-flat space-time.