Under the operation of connected sum, the set of knots become a monoid. The unknot is the identity, and associativity is given by the definition of connected sum. There is no way to find inverses on this set under the operation, so this is not a group.
In a lot of the texts I've read, authors seem to gloss over this fact, and move on to talk about using knot concordance and other methods to create a group structure.
Is this monoid really not useful at all? It seems quite a fundamental thing, so I thought that it must have use in classifying knots.
I also thought that the connected sum was quite a natural way to combine knots, because it is how we decompose into prime knots. However, the fact that you need to modify the operation so much in order to get a group structure makes me doubt this. It seems like there is a bit more going on than I initially assumed.
So my question is:
Is there any use for this monoid, and if not, does that mean that the connected sum isn't really the most natural operation on the set of knots?
Best Answer
The monoid structure is useful, but there's not much reason to study it per se because the structure is completely known: it's a free commutative monoid with countably many generators. There is one generator per prime knot (where the knots are oriented for connect sums to be well-defined).
There are a few components to this result. The first is that prime decompositions exist (this follows from additivity of Seifert genus). The second is that connect sum is commutative (there's a straightforward isotopy). The third is that prime decompositions are unique (in Lickorish "An Introduction to Knot Theory," this is summarized in Theorem 2.12). This unique representation result implies that the commutative monoid is freely generated by the prime knots, so it's isomorphic to $\bigoplus_{i} \mathbb{N}$ where $i$ ranges over prime knot types.
A consequence of all this is that the data of an oriented knot is simply the vector $(c_1,c_2,\dots)$ with each $c_i\in \mathbb{N}$, all but finitely many nonzero, where $c_i$ counts the number of times prime knot $i$ appears in the knot. This vector is additive under connect sums, and every such vector corresponds to a knot.
Connect sums are a relatively natural operation on knots. From the point of view of the knot exterior, a connect sum is evidenced by an annulus that meets the knot along a pair of meridians. Or, from another point of view, if the knot exterior is viewed as a ball with a knotted tunnel bored out, then you're gluing the knot exteriors together along an annulus at a mouth of each tunnel, creating a larger ball and a larger tunnel.
[Edit: I'm misremembering something in this paragraph and the way I describe it doesn't work for non-split links. Leaving it in case someone knows how to fix it.] By the way, there's also a notion of a commutative semiring of links. When you do the connect sum of two links, you take all the pairwise connect sums between components of each link, and so the number of components is multiplied. The disjoint union of links is the addition operation and the connect sum is the multiplication operation. The empty link is the additive identity and the unknot is the multiplicative identity. This is isomorphic to the commutative semiring of polynomials over $\mathbb{N}$ in a countably infinite number of variables (one per prime knot type).
In any case, the concordance group for knots is a quotient of the knot monoid.