Can any knot always be recovered from its diagram? Do diagrams always exist? In knot theory it's common to represent knots by projecting them onto planes, ensuring that all crossings that appear involve only two strands, and that these crossings are properly marked, usually by breaking the under-strand one to distinguish it.
I have two questions:


*

*Can this diagram always be done, that is, ensure that all crossings involve only two strands?

*Given a diagram in these conditions, is it always possible to recover the original knot?


The first seems trivial in the case of a finite amount of crossings, but I'm pretty new to knot theory so I don't know what happens in the general case.
The second one might be pretty complex, but a bibliographic reference would be enough, if possible.
EDIT: I'll clarify the second question: Since a knot diagram is obtained projecting in a plane, we lose one dimension and hence it is to be expected to lose information. When we specify the crossings, we recover a lot of this information by now knowing which strand goes above and which one goes below.
Clearly we don't recover all information, but, do we recover enough to be able to reconstruct an equivalent 3D knot (that is, up to isotopy) based only on the 2D diagram without ambiguity?
 A: The term "knot" can be confusingly used in two different contexts. In the first context, a knot is a (piecewise linear or smooth) embedding of the circle $S^1$ into three space (either $\Bbb{R}^3$ or $S^3$). Two knots $K$ and $J$ are said to be equivalent if there is an orientation preserving homeomorphism $h:\Bbb{R}^3 \to \Bbb{R}^3$ such that $h(K)=J$, or informally, the two knots are equivalent if one can be deformed or isotoped into the other. Ambient isotopy is an equivalence relation on knots defined this way.
In the second context, a knot is the equivalence class of embedded circles in $\Bbb{R}^3$ where the equivalence relation is ambient isotopy as above.
There is a one-to-one correspondence between equivalence classes of knot diagrams (where the equivalence relation is generated by Reidemeister moves) and the second context of knots above (so equivalence classes of embeddings of circles where the equivalence relation is ambient isotopy).
However, every knot diagram corresponds to infinitely many different embeddings of a circle into $\Bbb{R}^3$. If we view the knot diagram as lying in the $xy$-plane, then there are infinitely many ways to assign the $z$-coordinate so that the projection to the $xy$-plane is the same, and the crossing information remains the same.
One way to obtain a non-unique embedding of the circle in $\Bbb{R}^3$ is to consider the knot to be embedded in the $xy$-plane except near the crossings. Near the crossings, the embedding of the knot is given by "crossing bubbles" as below.

