I am currently informing myself about the basics of knot and link theory. I learnt pretty quickly about prime knots and a unique decomposition theorem for knots stating :
$\text{Thm: Every (isotopy class of) knot decomposes uniquely as a connected sum of prime knots.}$
I was wondering if a similar result held for $n$-stranded links.
First of all one would need to make the connected sum operation well-defined in the case of links. This gets done in my framework by considering $n$-stranded links as embeddings $f : {\mathbb{R}}^{\amalg n} \rightarrow \mathbb{R}^3$ who coincide outside of $[-1,1]^{\amalg n}$ with a fixed linear embedding sending the copies of $\mathbb{R}$ to lines parallel to the $x$-axis, uniformly dispatched around the $x$-axis. The figure in the URL below illustrates this definition. Now, the connected sum operation can be defined as the concatenation, and this coincides with the actual connected sum of knot when they are seen as $1$-stranded links. An illustration is again given in the picture when $n=2$.
An exact transcription of the theorem above is impossible since the operation $#$ now has some invertible elements, for example the one illustrated in the picture. But is there a reasonable subfamily of links for which it holds ? Or is there another operation for which one gets similar results ?
Thanks in advance !