By the way, there is a beautiful proof for the case of the unknot. I'll sketch it here (though I'll be a bit glib about technical issues). Assume that $X$ and $Y$ are knots and that $X \oplus Y = K$, where $K$ is the unknot (here I'm denoting the connect sum with $\oplus$). It makes perfect sense to take an infinite connect sum -- just keep shrinking the successive knots down closer and closer to a point. Of course, the result will be a wild knot, but that's no problem. Anyway, one can check that $K \oplus K \oplus \cdots$ is still the unknot. We can then do the following calculation.
$$K = K \oplus K \oplus \cdots = (X \oplus Y) \oplus (X \oplus Y) \oplus \cdots = X \oplus (Y \oplus X) \oplus (Y \oplus X) \oplus \cdots $$
$$= X \oplus K \oplus K \oplus \cdots = X.$$
More details for this are in the first chapter of Prasolov and Sossinsky's book on knot theory.