Does the monoid of knots form a ring? Consider the collection of all oriented knots modulo ambient isotopy. There is a composition operation which turns this collection into a monoid. I would like to know if there is a second operation --- an addition you might call it --- that turns the whole thing into a ring. Feel free to enlarge the set somewhat, if that's needed.
Some thoughts. Almost certainly you will want a slightly larger set to work in. Indeed what would the unit of addition be? It can't be the unknot, since that one's taken by composition. A more sensible choice would be the empty set. 
Well, is there an obvious choice of operation whose unit is the empty set? There is: rather than taking the set of knots, consider the set of links, and define the addition to be the unlinked disjoint union. Unfortunately I don't know if the knot composition can be extended to links in a well-defined manner, so I'm not sure if it's the right way to proceed.
Also before you ask, "Why should you expect a ring structure in the first place?" The answer is I don't know, I'm just curious if it can be done.
 A: One ring I've thought a little about is formal linear combinations of oriented knots over some ring (for example $\mathbb{Z}$), with multiplication being connect sum extended by linearity.  The existence of prime decompositions implies this is actually a polynomial ring over countably infinite generators, one per oriented prime knot. (So, for example, a generator for both the left-handed and the right-handed trefoil knots.)
The additive identity is $0$, the linear combination of no knots.  Beware that $2\langle\mathrm{trefoil}\rangle$ is not $\langle \mathrm{trefoil}\mathbin{\#}\mathrm{trefoil}\rangle$; the latter is $\langle\mathrm{trefoil}\rangle^2$, which is different.
Knot invariants that are multiplicative under connect sum would be quotients of this ring.  It is also possible to instead use disjoint union for the product and consider general links.  If you take this ring over $\mathbb{Z}[t^{\pm 1/2}]$ and quotient it by a particular skein relation, the quotient space is generated by the empty diagram and the unknot, and the Jones polynomial of a link is the polynomial coefficient of the unknot representative.
This sort of ring is used by Chmutov and Duzhin for connected $3$-regular graphs, which in their case they are interested in graph invariants coming from Lie algebras so they take the quotient by the IHX relation.
Duzhin, S. V.; Kaishev, A. I.; Chmutov, S. V., The algebra of 3-graphs, Proc. Steklov Inst. Math. 221, 157-186 (1998); translation from Tr. Mat. Inst. Steklova. 221, 168-196 (1998). ZBL0944.57009.
Another algebraic structure, which I have no idea the utility of, is a semiring of links -- in the vein of what you describe.  Addition is disjoint union, and multiplication is the disjoint union of the connect sum of each pair of components (in other words, we define multiplication on knots by connect sum and extend by "linearity" over disjoint union).  The additive identity is the empty diagram, and the multiplicative identity is the unknot.  This is an $\mathbb{N}$-algebra by having $n\in\mathbb{N}$ map to a disjoint union of $n$ copies of the unknot, which in other words means $n L$ is the disjoint union of $L$ with itself $n$ times.  Though, if you formally add in the additive inverses, you do get a ring isomorphic to the first one I mentioned over $\mathbb{Z}$.  This means the interpretation of $2\langle\mathrm{trefoil}\rangle$ would be $\langle\mathrm{trefoil}\amalg\mathrm{trefoil}\rangle$.
A: Formal sums would probably work (i.e. you just add knots algebraically without bothering too much about what it means geometrically). If you need a geometric intuition, one knot added to another would be the two knots, in space, not entangled in one another.
Then yes, the additive identity is the empty not.
This does mean that you need negative knots too, which again might not have a nice geometric interpretation (red negative knots vs positive black knots, perhaps?)
