Do knots behave the same way if they are defined to have "endpoints at infinity"?

Question

Typically, knots are defined as embeddings of $S^1$ into $\mathbb{R}^3$ or $S^3$. However, in real life, knots usually sit in the middle of a long rope, whose endpoints you might as well model as infinitely far apart from one another. This suggests an alternate definition of a knot as an embedding of $\mathbb{R}$ into $\mathbb{R}^3$, such that the embedding eventually goes off to infinity in two directions. Perhaps you could say that the embedding has to be a straight line except within a bounded region, or some such. My question is: does this definition differ from the ordinary one in any meaningful way? Intuitively, the ordinary definition would seem to be equivalent to taking a knot in the above sense and gluing the "ends at infinity" together. When manipulating an ordinary knot, we can freely pass a loop across this gluing point, whereas under the above definition of a knot we can't. But I'm not really sure if that makes any substantive difference to the theory. Are these definitions equivalent?