I'm trying to visualize the complement of the unknot embedded in $S^3$ but what I visualize isn't what I know the answer to be. Let $K$ be the unknot (a circle), here is what I've been told the answer is, $$S^3\setminus K\text{ is } S^1\times D^2$$ where $S^1\times D^2$ is the solid torus. But my intuition tells me that $S^1\times D^2$ should be $S^3\setminus S^2$. I'm certain I'm thinking about this the wrong way so I was hoping someone could explain to me where I've gone wrong.
Edit: Oh wait, in $S^3$ a circle in homotopic to a point .... did I answer my own question?