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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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3 Answers

up vote 2 down vote accepted

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



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Awesome, thanks. –  Owen Sizemore May 6 '10 at 20:25
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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
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This is more general than what you ask for, but the following paper by Ryan Budney is deeply relevant: JSJ-decompositions of knot and link complements in the 3-sphere. L'enseignement Mathe'matique (2) 52 (2006), 319--359 math/0506523.
By looking at the JSJ decomposition of knot complements, Ryan shows, among other things, that any knot can be constructed via "satellite operations" from hyperbolic knots and torus knots (in particular these are prime knots) in an essentially unique way. Connect-sums are an example of a satellite operation. As he mentions, this theorem (in some form) is also in an unpublished manuscript of Bonahon and Seibenmann; and Schubert proved some of it (is this right?). The paper is quite readable, and in English, and the result is much stronger than Schubert's Theorem.

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Bonahon and Siebenmann's preprint (30 years old now) has been put up on Francis Bonahon's webpage. I imagine it will be published in not too long. It's also a good reference for this information and covers the material quite similarly to the way I cover it. Bonahon and Siebenmann go further to explore the JSJ-decomposition of the Z_2-branched cover of S^3 branched over the knot or link, and spends less time on the "plain" JSJ-decomposition, for which Schubert's theorem falls out of. –  Ryan Budney May 13 '10 at 13:33
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