# Reference for a theorem on crossing changes of links

I've recently stumbled upon a paper of Scharlemann on crossing changes:

link text

In particular I am interested in understanding Theorem 2.2 (page 6):

"Theorem: If links A and B are related by a crossing change, and both are composite, then the crossing change takes place within a proper summand."

Where can I find a proof of this result? Also, is it possible to find an equivalent statement on diagrams?

Thanks

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## 1 Answer

The reference is given in the paper you cite: M. Eudave-Mun ̃oz, Primeness and sums of tangles, Trans. Am. Math. Soc. 306, 773-790 (1988)

The arguments are purely combinatorial, but there should be a simplification using sutured manifold theory along the lines of Scharlemann and Thompson's paper "Unknotting number, genus, and companion tori" by Scharlemann and Thompson MR0929535

Off the top of my head, I don't see a way to convert either type of argument into a something about diagrams.

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The results in the cited paper are about rational tangles, while the statement in Scharlemann's paper is about summands of a link. I can't really figure out how to proof Theorem 2.2 starting from the paper of Eudave-Mun ̃oz. –  Springfield Jun 14 '13 at 10:09
A crossing change is rational tangle replacement of distance 2. If you remove a regular neighborhood of the crossing you get a tangle $(B,t)$. Putting the crossing in one way produces a knot $k_1$ and putting it in the other way (with the crossing reversed) produces a knot $k_2$. The rational tangles are the intersections of $k_1$ and $k_2$ with the complement of $B$. Each of those rational tangles has a disc with boundary in the four-punctured sphere $\partial B - t$. The discs can be isotoped to intersect in two arcs, and so the distance between the rational tangles is 2 –  Scott Taylor Jun 16 '13 at 20:34
(continued) Assume that $k_1$ and $k_2$ are composite. By Theorem 1 from the Eudave-Munoz paper the tangle $(B,t)$ cannot be prime. By the definition of prime tangle either this means that a strand of $(B,t)$ has a local knot (in which case, you can show the crossing change occurs in a factor of $k_1$ and $k_2$) or the tangle $(B,t)$ is rational (in which case, $k_1$ and $k_2$ are 2-bridge and so can't be composite). –  Scott Taylor Jun 16 '13 at 20:39