# A Reference for Schubert's Theorem

Schubert's Theorem in Knot Theory says that any knot can be uniquely decomposed as the connected sum of prime knots.

Unfortunately the original paper is in German.

Does anyone know a good english reference for this. Or just the special case of the unknot. (i.e. that the unknot can't be written as the connected sum of two knots which aren't the unknot.)

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A fairly standard reference would be "Knot theory" by G.Burde and H. Zieschang, Chapter 7.

Roland

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Awesome, thanks. –  Owen Sizemore May 6 '10 at 20:25

By the way, there is a beautiful proof for the case of the unknot. I'll sketch it here (though I'll be a bit glib about technical issues). Assume that $X$ and $Y$ are knots and that $X \oplus Y = K$, where $K$ is the unknot (here I'm denoting the connect sum with $\oplus$). It makes perfect sense to take an infinite connect sum -- just keep shrinking the successive knots down closer and closer to a point. Of course, the result will be a wild knot, but that's no problem. Anyway, one can check that $K \oplus K \oplus \cdots$ is still the unknot. We can then do the following calculation.

$$K = K \oplus K \oplus \cdots = (X \oplus Y) \oplus (X \oplus Y) \oplus \cdots = X \oplus (Y \oplus X) \oplus (Y \oplus X) \oplus \cdots$$ $$= X \oplus K \oplus K \oplus \cdots = X.$$

More details for this are in the first chapter of Prasolov and Sossinsky's book on knot theory.

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Great! That is very elegant. –  Owen Sizemore May 6 '10 at 20:59
Eilenberg's swindle strikes again! –  Graham Leuschke May 6 '10 at 23:40
Yep! It's a great trick in topology. For a related result due to Barry Mazur (back when he was a grad student and hadn't yet abandoned us simple topologists to become a number theorist), see the following anecdote of Stallings : math.berkeley.edu/~stall/mazur.html –  Andy Putman May 6 '10 at 23:48
And Mazur reminisces take on grad school here: math.harvard.edu/~mazur/remembrances/john_stallings.pdf –  j.c. May 8 '10 at 18:56