Q:

Beginnings of Topology: Homeomorphisms

Why is a knot and a circle homeomorphic? The general definition of a homeomorphism requires that you be able to deform each to one another.

A:

You are confusing homeomorphism, a bijection that is in both directions continuous, and homotopy, which is a deformation from one form to the other. 
In the case of knots, one needs ambient homotopies, and only the trivial knots are, by definition, ambient homotopic to the circle.

Added: You can of course deform every simple knot into a circle. However, this most often requires self-intersections, which one wants to exclude in knot theory. That is the reason why "ambient". For a knot deformation one requires that a tube around the knot is also moved along, and that at all stages of the homotopy the parametrization of the tube is, locally as well as globally, injective. No folds and no self-intersections allowed.

