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.)


1 Answer 1


For the fact that the connect sum of alternating knots is alternating, you can simplify the argument a little bit by noting that one of the following two diagrams will be alternating:

Connect sums

For the converse, that is a result of Menasco. See Theorem 4.4 of Lickorish's "An introduction to knot theory."

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.


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