# A knot $K = K_1 \# K_2$ is alternating if and only if $K_1$ and $K_2$ are alternating.

Is a knot $$K = K_1 \# K_2$$ is alternating if and only if $$K_1$$ and $$K_2$$ are alternating?

In particular, I'm interested in the following direction:

If $$K_1 + K_2$$ is an alternating knot, are both $$K_1$$ and $$K_2$$ alternating?

I was able to prove the other direction. An outline is as follows: suppose $$K_1$$ and $$K_2$$ are alternating, and consider alternating diagrams for them. Each segment on the outer edge of the diagrams viewed as subsets of $$\mathbb R^2$$ inherits two orientations - one from the embedding in $$\mathbb R^2$$, and one from the "under-crossing/over-crossing" data. Therefore each outer-edge segment can be given a sign $$\pm$$ depending on whether the inherited orientations agree. If both $$D_1$$ and $$D_2$$ have outer-edge segments with the same sign, we can take the connected sum along a path joining these two segments to obtain an alternating diagram for $$K_1 + K_2$$. If all of the outer-edge segments of $$D_1$$ have opposite sign to those of $$D_2$$, I was able to construct an isotopy which gives a new alternating diagram $$D_2'$$ which changes the sign of a given segment, reducing this to the previous case. (The isotopy is a bit like rotating a solid torus along a meridian by 180 degrees, given a suitably chosen embedding of the knot inside the solid torus.)

Theorem 4.4. Suppose $$L$$ is a link that has an alternating diagram $$D$$. Then $$L$$ is a prime link if and only if $$D$$ is a prime diagram.
Hence, if $$K_1\mathbin{\#}K_2$$ is an alternating knot, it has an alternating diagram, and the diagram will have a connect sum decomposition loops. This circle might not be for the given connect sum, but by induction there will be a collection of connect sum decomposition loops that decomposes the knot into a connect sum of prime knots, each of which is alternating because the diagram gives alternating diagrams for each of them. Hence, $$K_1$$ and $$K_2$$ must both be alternating, since they are connect sums of alternating knots.