In Knots and Links by Cromwell he states that is it conjectured that this upper bound of the unknotting number of a composite link is in fact an equality
$$u(L_1\#L_2) \leq u(L_1) + u(L_2) $$
I am trying to understand this conjecture. Correct me if I'm wrong:
Given any diagram $D_1$ of $L_1$ and $D_2$ of $L_2$ we know that this holds: $$u(D_1\# D_2) = u(D_1) + u(D_2)$$ The problem is to prove that for any other diagram $D_3$ of $L_1\# L_2$ $$u(D_3) \geq u(D_1 \# D_2)$$
Is this the problem with proving this conjecture? If not can you explain what the problem is?
There's no need to bring diagrams into it: imagine unknotting $L_i$ with a minimal number of crossing changes (which for a link, I suppose you mean to end up with a completely split link of unknots). The data can be recorded as a collection of arcs (edit: actually bands; I'll fix this later) in the complement of $L_i$ that are incident to the link, where the unknotting can be read off by taking a small patch of the link and isotoping it along the arc, and the patch ultimately is passed through the patch of link at the other end of the arc.
If you take some connect sum $L_1\#L_2$ (which, recall, is not well-defined for links without saying exactly where the connect sum occurs), you can bring over the unknotting arcs as well. If you do the unknotting operations along these arcs, $L_1\#L_2$ is certainly a split link of unknots. (There is a sense in which connect sum and unknotting in a region away from the connect sum site are commutative.) This means $u(L_1\#L_2)\leq u(L_1)+u(L_2)$ because one could imagine there might be a smaller collection of arcs for $L_1\#L_2$ itself. For instance, an unknotting arc might go between $L_1$ and $L_2$ in the connect sum.
If you do want to think about diagrams, first of all let's make sure $D_1$ and $D_2$ are diagrams that represent the unknotting numbers. By that, I mean $u(D_i)=u(L_i)$, where $u(D_i)$ I presume is the number of crossings in the diagram to flip so that it becomes a diagram for a split link of unknots. By the above reasoning, we have $u(D_1\#D_2)\leq u(D_1)+u(D_2)$. For diagrams -- where $D_1\#D_2$ has essentially the same crossings as $D_1$ and $D_2$ -- this is an equality, basically because knots under connect sum have no additive inverses, so an unknotting of the diagram $D_1\#D_2$ corresponds to unknottings of $D_1$ and $D_2$.
By the way, any collection of arcs can be represented as crossings in some diagram. First isotope the link so that the arcs become vanishingly small, then use the usual diagram projection techniques. This correspondence implies that $u(L)$ for a link $L$ is the minimum of all $u(D)$ for $D$ ranging over diagrams for $L$.
Since we made sure $u(D_1)+u(D_2)=u(L_1)+u(L_2)$, if one were to manage to show that for any diagram $D_3$ of $L_1\#L_2$ that $u(D_3)\geq u(D_1\#D_2)$, then $u(L_1\#L_2)\geq u(L_1)+u(L_2)$, completing the proof of the conjecture.
In a logical sense, this is the problem with the conjecture. However, it is unclear to me whether talking about diagrams is the right way to go. For instance, you will have to somehow relate $D_3$ to $D_1$ and $D_2$.