Two knots $K$ and $K^\prime$ are equivalent if and only if their projections $P(K)$ and $P(K^\prime )$ are equivalent I have been trying to find a proof for this theorem online but can't seem to find one anywhere, and am unsure where to start in terms of trying to write my own proof.
I think what is mostly confusing me is whether Reidmeister moves and elementary knot moves are technically applied to the knot or the projection, as I would assume I would need to start with the theorem that $K$ and $K^\prime$ are equivalent if and only if $K^\prime$ can be obtained from $K$ by applying a finite number of elementary knot moves.
However, are we not technically doing this to the projection? Or I guess we can say we are doing it to both?
If anyone has any tips or can point me in the direction of a proof online I would be very grateful.
 A: This is known as Reidemeister's theorem. The Reidemeister moves apply to the projections and not the knot; they let you work with just the combinatorial data of a knot diagram.  The easy part of the theorem is that each Reidemeister move for a projection can be interpreted as an isotopy of the corresponding knot, and the harder part is that if two knots are equivalent, then there exists a sequence of Reidemeister moves bringing the first projection to the second.
You might look at Reidemeister's original 1927 paper to see how he did it (there's a very brief overview at the beginning of Lickorish's "An Introduction to Knot Theory"). One thing to note about low-dimensional topology is that you have to make a choice between working with smooth topology or piecewise-linear topology. Reidemeister, and pretty much every other author, has worked with the piecewise-linear case. In a way it doesn't matter which you choose, since there's a theorem that the categories of smooth and PL manifolds are equivalent in some sense. However, it's not so clear how you pass from one setting to the other (or at least I've never gone through the details myself!)
If you want to prove Reidemeister's theorem in the smooth setting, there's Roseman's papers (see papers with "Reidemeister" in their title and references), but they do seem to require some familiarity with singularity theory that's not explained -- mainly it's using a result that smooth maps in certain dimensions can be perturbed to only have Morin singularities, which here have particularly simple normal forms locally described by polynomials up to only the third degree. In some of my work, there's a part where I describe how to derive Reidemeister's theorem in the smooth setting using singularity theory via a variation on the Thom-Boardman stratification of jet spaces, which I hope to be able to make available soon.
A related problem is considering, rather than knots, just smooth maps $S^1\to \mathbb{R}^2$. It turns out every smooth isotopy $F:S^1\times\mathbb{R}\to\mathbb{R}^2$ can be perturbed so that for all but finitely many values of $t$ the map $F_t:S^1\to\mathbb{R}^2$ is a smooth immersion with normal crossings (in this case, no three points map to the same point, and when two points map to the same point they do so such that their tangent vectors span a 2D space), and otherwise it has exactly one of the following three singularities:

The first singularity can be locally modeled by $F_t(x) = (x^3+xt, x^2)$, the second by parabolas moving in time, and the third by three lines moving in time.
