Equivalence of knots It's intuitively clear what it means that two knots $K,K'$ are essentially the same, but it can be termed and defined more precisely in different ways. Are all of them equivalent?


*

*$K, K'$ are homotopy equivalent (or of the same homotopy type)

*$K, K'$ are isotopic 

*$K, K'$ are related by an ambient isotopy
 A: Isotopy (as opposed to ambient isotopy) is not a good equivalence relation on knots, because any knot is isotopic to an unknot.  The idea is to pull the knot tighter and tighter until it shrinks to a point.  More precisely (but still not completely precisely, because that would be a mess to write down), the isotopy would start with the given knot at time $t=0$.  At a later time $t$ (in $(0,1)$), it would look like a $t$ fraction of a circle (unknotted) with a copy of the original knot compressed into a $(1-t)$-sized part.  Finally, at time $t=0$, it's the whole unknotted circle.
This is why people like to define equivalence of knots to mean ambient isotopy rather than mere isotopy.
To the best of my knowledge, there is an equivalent (to ambient isotopy) definition that says essentially that there is a self-homeomorphism of the ambient space sending the one knot to the other.  Here, "essentially" refers to the fact that one has to be careful about orientations; it seems the self-homeomorphism of the ambient space should preserve orientation, but maybe there's a subtler point here that I'm overlooking.
