$\mathbb{R}^3$ and $S^3$ in the Definition of a Knot Definition: A knot $K$ is a smooth embedding of a circle $S^1$ into 3-dim Euclidean space $\mathbb{R}^3$ (or the 3-dim sphere $S^3$).
What confuses me about this definition is that as far as I know, $\mathbb{R}^3$ and $S^3$ are different from each other. Isn't $S^3$ embedded into 4-dim Euclidean space?
If you can please explain to me the reason why $\mathbb{R}^3$ and $S^3$ are used interchangeably in this definition, I would appreciate it. Also, please consider that I didn't take a course on topology, unfortunately.
 A: Fundamentally there is little real importance as to which space you use until you start doing some hard-core topology.
[An example is when one studies embedded surfaces in 4-manifolds with boundary. The the (connected components of the) boundary of the 4-manifold is often a 3-sphere or a 3-torus, and the boundary of the embedded surface will be a knot (or link) in this space]
One nice reason for studying knots in $S^3$ rather than $\mathbb{R}^3$ is that two inequivalent ways of defining a knot in $\mathbb{R}^3$ become equivalent in $S^3$. In particular, the every-day notion of a knot does not work as a mathematical definition since such a knot is made of a piece of string with loose ends. Mathematically these loose ends mean that any knot can be "untied" by an ambient isotopy of $\mathbb{R}^3$ so all knots are topologically the same.
There are two ways this is often fixed. The first is the one which you find in any standard definition of a knot: you glue the ends of the string together to form a circle. Then a smooth embedding of this circle into $\mathbb{R}^3$ cannot usually be untied by an ambient isotopy, leading to some interesting topology.

The other alternative definition is to say that the two endpoints of the string stretch off to infinity in two different directions. An ambient isotopy cannot pull these ends away from infinity, so again most knots cannot be untied.

Using stereographic projection (or some other means) one can show that $S^3$ can be obtained from $\mathbb{R}^3$ by adding a single "point at infinity" - or equivalently, by removing a single point from $S^3$ you are left with a space homeomorphic to $\mathbb{R}^3$. Now consider the second definition of a knot. Here the ends of the string are sent to infinity. By adding a point at infinity these endpoints are identifies in $S^3$ and so the string closes up into a circle.
In this way the two different definitions end up coinciding in $S^3$.
