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.

http://arxiv.org/abs/1308.1594

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