Consider a two-component tame link in 3-space, consisting of an arc from $(-1,1,0)$ to $(1,1,0)$ and an arc from $(-1,-1,0)$ to $(1,-1,0)$, confined to the slab $-1 \leq x \leq 1$. Call such a link trivial if it can be deformed so that the arcs admit parameterizations in which the $x$-coordinate is strictly increasing. (Note that this notion of triviality is rather forgiving, in that it permits the arcs to twist around one another.) There is a natural way to add two such links by sticking them side-by-side. (More formally, one compresses the links so that the $x$-coordinate in the arcs goes from $-1$ to 0 and from 0 to 1, respectively, and then applies the usual way of composing paths to the respective arcs.) For an example of what such a sum can look like, see the top figure at <http://en.wikipedia.org/wiki/File:Fisherman>'s\_knot.png . [I can't get the link to embed properly; can anyone fix this?]


Can the sum of two non-trivial links of this kind be trivial?


Note the confinement property that prevents us from pulling the arcs outside the slab.


It is a classical result that the analogously-defined sum of two non-trivial 1-component links cannot be trivial. (Can anyone provide a definitive reference for this result? I learned about it from a Martin Gardner column that presented a proof due to Conway.)



Cancellation of 2-component links
What you are talking about are called *string links*. String links have a stacking monoid operation. It's non-commutative (provided you have two or more strands). The invertible elements are precisely the string links that are braids. There are a few references on this topic. 


I forget who first proved the result about invertible string links, but you should be able to find most of the references in this Blair-Burke-Koytcheff paper. They cite Krebes, but it might go back further. 


In the one-stranded case your monoid is free commutative with countably-infinite number of generators. That's Schubert's theorem (1949 *Die eindeutige Zerlegbarkeit eines Knotens in Primeknoten*). 


<http://arxiv.org/abs/1308.1594>


