The way knot theorists like to think about knots is as an embedding of a circle in three dimensions, considered up to deformations such that the circle doesn't break or pass through itself. The different classes of such configurations are different knots. Then there is an operation one can perform on two knots where you cut them both open and connect the loose ends together, forming a larger knot (actually, you need to specify some more information to make this procedure well-defined, but this won't matter for this discussion). Then the question you are asking is, are there two non-trivial knots, A and B, such that their sum, A+B, is the trivial knot, or "unknot"?

The answer to this is no, and there are a couple ways to prove it, but they are a bit technical, and I won't be able to give all the details. The first is more visual, but also more fishy-sounding, and goes as follows: we can actually form an infinite sum of knots, A1+A2+..., by taking the embeddings of the knots smaller and smaller so they all "fit" on a single knot. Then, suppose it were the case that A+B=0 (the unknot), and let us consider A+B+A+B+A+B+... On the one hand, if we group terms like (A+B)+(A+B)+..., this is clearly 0, but if we group them as A+(B+A)+(B+A)+..., it is A. Thus A=0. You might find this fishy since if I replace the knot equation A+B=0 by the real number equation 1+(-1)=0 and ran the same kind of argument again, I'd (incorrectly) prove 1=0. The reason is that such a regrouping of terms is invalid for an infinite sum of real numbers. But it turns out to be ok for the knot sum (this takes some more work to prove), and so A+B=0 is only possible if A (and so also B) are themselves the unknot.

Another, more technical proof, uses knot invariants. A knot invariant is some data, eg, an integer or polynomial, which one can assign to a knot which does not depend on how the knot is embedded. Thus one can use it to check if two knots are the same or not. One example is called the "knot genus", and is an integer one can assign to a knot. It would be too big of an excursion to describe how it is defined, but the important properties for us are:

1. The unknot has knot genus 0.
2. All non-trivial knots have knot genus greater than zero.
3. The knot genus of A+B is the sum of those of A and B.

From this it's easy to see that it's impossible for A+B to be the unknot.