There is a refinement of the connect-sum decomposition called <a href="http://en.wikipedia.org/wiki/Satellite_knot" title="satellite operations">'satellite operation'</a> or 'splicing'.  Like the connect-sum operation, it's originally due to Schubert, although Schubert didn't prove a uniqueness theorem.  That came much later with the JSJ-decomposition. 

Unfortunately, this consists of infinitely-many operations.  You could think of it as *one* operation that takes as input $n$ knots and an $(n+1)$-component link $L = (L_0, L_1, \cdots, L_n)$ such that the sublink $(L_1, \cdots, L_n)$ is trivial.  You also demand that the complement of $L$ in $S^3$ is either Seifert-fibred or hyperbolic.   It's also an essentially unique decomposition (like the connect-sum decomposition) -- the link is well defined but the indexing of the knots is only unique up to the symmetry group of the link $L$.  

![Whitehead double of figure-8][1]

This is called the Whitehead double of a figure-8 knot.  The link $W$ (in the $\mathbb G_K$ diagram) is the Whitehead link, the input knot (only one) is the figure-8 knot. 

For these definitions there is a framing convention, twisted Whitehead doubles would decompose as:

![Twisted double][2]

<img src="https://upload.wikimedia.org/wikipedia/en/7/7c/Knot_with_borromean_rings_in_jsj_decomp.png">

This doesn't have a standard name.  I like to call it a "(twisted) Borromean double".  The input link is the Borromean rings (technically, a twisted version of them), and the input knots are two trefoils.  Another example:

![Another twisted example][3]

This decomposition also applies to links in $S^3$.  Here is a picture of the decomposition in an interesting case: 

![Link decomp][4]


  [1]: https://i.sstatic.net/YEJpC.png
  [2]: https://i.sstatic.net/6l4oW.png
  [3]: https://i.sstatic.net/MyWvH.png
  [4]: https://i.sstatic.net/5pqKm.png