Suppose you have an arc in a ball with ends fixed on a boundary. In other words, we have an isolated knot on one of the arcs of a tangle. If we glue the ends we get an ordinary knot and clearly isotopic isolated knots, when ends are connected, give the same knot. But is this a bijection, i.e. can we have different knots of the first kind that give the same knot after the ends are glued?
This seems like it should be obvious, but I did try to google and ask people in my department - neither helped.