Classification of links with unknot components Question
I am interested in the links in $\mathbb{R}^3$ with trivial components. More precisely, I'd like to know if the classification of links with finitely many components, which are all unknots have been done.
Background
With only one component, the problem is trivial. With two components, I believe it should be classified by how many times they "wrap together" (linking number?). With three components, however, I fail to be confident to say I can classify all of them. I hope the classification can be done by drawing graphs.
Variants
It would also be nice to know of the variants:

*

*links -> oriented links

*unrestricted linking number -> each linking number is $\pm 1$.

 A: One of the oldest examples of nontrivial links with zero linking number is the Whitehead Link. 

There are many other examples. 
If your wish is to classify, say, 2-component, links where each component is an unknot, by some simple numerical invariant (like the linking number), then there is no such classification. (One can even make this statement precise once you have defined what "simple" is.) There is a classification, of sorts, which works for general links. It is given by Thurston's Hyperbolization Theorem. For instance, for 2-component links where both components are unknots, the classification says that either the link complement is hyperbolic or it is a torus link (the Hopf link is a simple example), or it is the result of a "satellite construction" (the complement contains an "essential torus"). This classification is quite useful but I am pretty sure, this is not what you asked for. As I said in my comments, if you are interested in knots and links, pick up a book (say, "Knots and Links" by Dale Rolfsen -- it is a bit dated but still covers all the basics) and start reading.    
