Definition of equivalence of knots I've just started reading An Introduction to Knot Theory from W. B. Raymond Lickorish and I've come up with the following definition:
Definition. Two knots $K_1$ and $K_2$ are equivalent if there exists an orientation-preserving PL-homeomorphism $h: S^3 \to S^3$ such that $h(L_1) = L_2$.
I can't understand why this definition is equivalent to the two knots being isotopic. Can anyone help me? Why does it need to be orientation-preserving? Would it work if it were $h: \mathbb{R}^3 \to \mathbb{R}^3$?
 A: This definition is relying on a key fact about PL (or smooth) topology: if $h: S^3 \to S^3$ is an orientation-preserving PL homeomorphism, then there is an isotopy $H : [0,1]\times S^3\to S^3$ such that $H_0=\operatorname{id}_{S^3}$ and $H_1=h$.  This is because the mapping class group of $S^3$ is trivial.  Since $h(L_1)=L_2$, then $H_t|_{L_1}:L_1\to S^3$ is an isotopy from $L_1$ to $L_2$ through PL embeddings.
The unrestricted $H$ is known as an ambient isotopy.  What you want from a definition of isotopy of knots is isotopy extension to ambient isotopies.  Intuitively, dragging the knots around should extend to dragging around the ambient space, too.  Why is this? You want any sorts of peripheral structures, like Seifert surfaces, to be able to follow along the isotopy, too.  If you have a continuous family $h:[0,1]\times S^1 \to S^3$ of PL embeddings, then this does indeed extend to an ambient isotopy.  And since the mapping class group is trivial, the only data you need out of this is the single orientation-preserving PL homeomorphism of $S^3$ that carries the knot to the end result of the isotopy.
There is a strange detail in here: while $h:S^3\to S^3$ does come from an ambient isotopy, there can be many ambient isotopies it comes from that are not isotopic to each other (yes, non-isotopic isotopies :-)).  This can happen when a knot is a connect sum: a connect sum of two right-handed trefoil knots has an isotopy that swaps the two connect summands, and this isotopy should be non-isotopic to the identity isotopy.  This detail does not matter for the definition of knot equivalence, though.
