TITLE: Computing the structure of the group completion of an abelian monoid, how hard can it be?
QUESTION [11 upvotes]: Cherry Kearton, Bayer-Fluckiger and others have results that say the monoid of isotopy classes of smooth oriented embeddings of $S^n$ in $S^{n+2}$ is not a free commutative monoid provided $n \geq 3$.   The monoid structure I'm referring to is the connect sum of knots.
Bayer-Fluckiger has a result in particular that says you can satisfy these equations $$a+b=a+c, \ \ \ \  b \neq c$$ 
where $a,b,c$ are isotopy classes of knots and $+$ is connect sum.
When $n=1$ it's an old result of Horst Schubert's that the monoid of knots is free commutative on countably-infinite many generators. 
What I'm wondering is, does anyone have an idea of how difficult it might be to compute the structure of the group completion of the monoid of knots, say, for $n \geq 3$?  That's not really my question for the forum, though.  
It's this: Do people have good examples where it's "easy" to compute the group-completion of a commutative monoid, but for which the monoid itself is still rather mysterious?  Meaning, one where rather minimal amounts of information are required to compute the group completion?  Presumably there are examples where it's painfully difficult to say anything about the group completion?  For example, can it be hard to say if there's torsion in the group completion?

REPLY [10 votes]: An ultra-classical example: the failure of unique factorization in algebraic number fields. Here one looks at the multiplicative monoid of nonzero algebraic integers in a finite extension field of $\mathbb Q$. Factoring out units gives a quotient monoid $M$, and this is free (abelian) on the irreducible elements exactly when the unique factorization property holds. The monoid $M$ embeds in the ideal group, the free abelian group generated by the prime ideals, and the group completion of $M$ is the finite-index subgroup of the ideal group generated by the principal ideals. This subgroup is also free, but without a canonical basis. One can think of this subgroup as a somewhat skewed lattice in the ideal group, and $M$ is the intersection of this lattice with the positive "orthant" of the ideal group. 
I'm not an expert on this stuff, so please correct any inaccuracies in what I said above. I gather that the structure of $M$ as a monoid can be rather complicated, even though it's a submonoid of a free monoid.  Perhaps this complexity is why the monoid structure seems rarely to be discussed explicitly. Does anyone know any references describing the monoid structure?