Actually, several years ago I was in a short, introductory, course about knot theory, and my original question that I posed was: "can the knots be used to classify homeomorphims in $\mathbb{R}^3$?". What I was aiming at, concerned both to see about a proof if there were enough knots to cover the whole of $\mathbb{R}^3$ and if it is the case, whether knots could be used as a sort of method to classify homeomorphisms.
Going deeper into knot theory, homeomorphisms are not enough to clearly distinguish among knots. Not even homotopies, one having to use actually ambient isotopies to really find mappings which would tell one knot from another; that is, if I understood the definitions correctly.
There are other definitions that I have found, but still I am not entirely sure on how to use them to answer my questions. Here they are, with their source:
"Classification theorem for closed surfaces 3.33: A closed, path-connected surface is homeomorphic to a 2-sphere, a connected sum of tori, or a connected sum of projective planes. The Euler characteristic and orientability of the surface distinguish among these possibilities" [Robert Messer, Philip Straffin, "Topology Now!, The Mathematical Association of America]
"Two smooth knots $K_0$ and $K_1$ are called equivalent if there exists a one-parameter family $f_t:\mathbb{R}^3\rightarrow \mathbb{R}^3, r \in [0,1]$, of diffeomorphisms smoothly depending on the parameter $t$, taking the knot $K_0$ to the knot $K_1$, i.e., such that $f_0$ is the identity and $f_1(K_0) = K_1$. (Here 'smoothly' means that the map $F: \mathbb{R}^3 \times [0,1] \rightarrow \mathbb{R}^3 $ given by $(x,t) \mapsto f_t(x)$ is differentiable.)The familiy of diffeomorphisms $f_t$ is said to be an isotopy joining the knots $K_0$ and $K_1$; for this reason equivalent knots are also called isotopic or ambient isotopic" [V. V. Prasolov, A. B. Sossinsky, "Knots, Links, Braid and 3-Manifolds. An Introduction to the New Invariants in Low-Dimensional Topology", American Mathematical Society]
"(2.1.1) Theorem. Two knots are ambient isotopic if and only if any diagram of one is equivalent to any diagram of the other under Reidemeister moves" [Dominic J. A. Welsh, "Complexity. Knots, Colourings and Counting", Cambridge University Press]
"Ambient isotopy is an equivalence relation on knots: two knots are equivalent if they can be deformed into one another. Each equivalence class of knots is called a knot type; equivalent knots have the same type." [Peter R. Cromwell, "Knots and Links", Cambridge University Press]
All the aforementioned statements say something about equivalence of knots (and suggest how to distinguish among them if there is a doubt), but, with the exception of the statement taken from Prasolov and Sossinsky, they do not really say much about its relationship with $\mathbb{R}^3$, and the questions that I posed, concerning the amount of coverage of knots of $\mathbb{R}^3$ and the classification of at least some of its subsets.