Q:

Computing the structure of the group completion of an abelian monoid, how hard can it be?

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?

A:

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?

This happens all the time in K-theory $K^0(X)$, both algebraic and topological.  Perhaps it is even the reason that K-theory is a useful tool.
For a striking algebraic example, take $X = \mathbb{A}^n_k$ where $k$ is a field.  Then $K^0(X)$ is the group completion of the commutative monoid $M$ of isomorphism classes of finitely generated projective modules over $R = k[x_1, \ldots, x_n]$.  In 1955 Serre asked whether every such module was free, i.e., whether $M = \mathbb{N}$.  This question became known as Serre's conjecture.  Serre proved in 1957 that every finitely generated projective $R$-module is stably free, i.e., $K^0(X) = \mathbb{Z}$.  However, it was not until 1976 that Quillen and Suslin independently proved Serre's original conjecture.  So between 1957 and 1976, $M$ was an example of a commutative monoid whose group completion was known but which itself was not known.  This is only a historical example, because $M = \mathbb{N}$ turns out to be very simple; however, it illustrates the difficulty of the question in general.
A topological example where the commutative monoid is not so simple is given by $KO^0(S^n)$.  Let us take $n$ congruent to 3, 5, 6, or 7 modulo 8, so that $KO^0(S^n) = \mathbb{Z}$ by Bott periodicity (the generator being given by the trivial one-dimensional real vector bundle).  Let $T$ be the tangent bundle to $S^n$.  In $KO^0(S^n)$, of course, the class of $T$ is equal to its dimension $n$.  But if we let $M$ be the commutative monoid of isomorphism classes of finite-dimensional real vector bundles on $S^n$ (so that $KO^0(S^n)$ is the group completion of $M$) then the class of $T$ is not equal to the class of the trivial $n$-dimensional vector bundle unless $S^n$ is parallelizable, which only happens when $n$ is equal to (0 or 1 or) 3 or 7.  So for all other values of $n$, $M$ is not simply $\mathbb{N}$; there are extra vector bundles which get killed by the group completion process.  Understanding these monoids $M$ for all $n$ amounts to understanding the homotopy groups of all the groups $O(m)$, which I expect is not much easier than understanding unstable homotopy groups of spheres.
Finally, Pete's example of the monoid of cardinalities of at most countable sets and its absorbing element also makes an appearance in K-theory; here it is called the Eilenberg swindle and it explains why we restrict ourselves to finitely-generated projective modules.

A:

This example is similar to Reid's: If $G$ is a finite group, then $K^0(G-\mathrm{rep})$ is just the class functions on $G$. But the question of whether a specific class function is the character of a representation, or only of a virtual representation, can be very hard. In a sense, Mark Haiman got tenure at Berkeley for proving that certain class functions on $S_n$ were characters.

A:

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?

