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Let $L=K_1 \cup K_2$ be a two-components link in a copy of $S^3$ and let $K$ be a knot, thought in a different copy of $S^3$. In other words, we have two couples $(S^3, L)$ and $(S^3, K)$. Let us define a notion of "connected sum" between objects of this kind: we choose a component of $L$ (for example $K_1$) and two arcs $S \subset K_1$ and $S' \subset K$. We define the connected sum of the couples $(S^3, L)$ and $(S^3, K)$ along $K_1$ as the object $(S^3 \# S^3 \simeq S^3, K \# K_1 \cup K_2)$, where the connected sum of the two copies of $S^3$ is performed along disks which intersect the links in the chosen intervals $S$ and $S'$.

I am not sure that this operation is well defined; it would be nice if it was but I wouldn't bet on that. What I would like to prove is that if $L$ is a split link, then the result of this operation is again a split link. The problems seems to be the fact that the bars used to perform the connected sum of knots could intersect the other component ($K_2$); I think that it could be possible to perform the connected sum of knots "inside" the disks used to perform the connected sum of the two copies of $S^3$. Is it correct? Would it be enough to state the the resulting link is split, under the hypothesis that $L$ is split? Thanks in advance.

P.S. I found this definition of connected sum of links in "Three-dimensional link theory and Invariants of Plane Curve singularities" by Eisenbud & Neumann. This definition can be in fact easily generalized to the connected sums of two links, but things may turn to be more complicated in that case.

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As stated this is not a correct definition of the connected sum of two knots/links. This is called a 'band sum'. The band sum is not well-defined: you can obtain many different knots as a result of the band sum of two given knots. (Try to picture this by imagining your band winding around one of the knots a few times before actually connecting up with it.)

The correct definition demands that there be a regular planar projection of the knots so that the projection of this disc only intersects the projection of the knots in the arcs you have already specified.

That is, take two knot diagrams, and do your connect-sum by attaching a band which only intersects the knot diagrams in those arcs. This prevents the "winding" I did above.

I believe but have not checked that the definition you get from this is well-defined (after specifying the component of L, of course); that this is well-defined for knots ought to be in any half-decent knot theory textbook (say Rolfsen). And it should be easy to see that the resulting link is indeed split (choose a diagram so that L is split by a line in the plane, then take K to be on one side of that line and your band to be on one side of that line).

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  • $\begingroup$ It seems that I have only to check that this construction does not depend on the diagram (but this should not be difficult, using Reidmeister moves I can always obtain a split diagram) and that I can take $K$ to be on one side of the line...for this second part can I use an argument like "all disks are isotopic"? So that I can choose one such disk in $S^3$ so that the projection stays on one side of the line? I hope I was clear. $\endgroup$ Feb 18, 2021 at 15:29
  • $\begingroup$ I would also appreciate if you can give me a reference for a precise definition of these link constructions. The book by Eisenbud and Neumann has other purposes and does not focalizes much on that. $\endgroup$ Feb 18, 2021 at 15:31

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