I'm trying to fill a woeful gap in my topological knowledge and learn a little knot and link theory (I'll be recording my progress on the nLab, starting with a page on links). Not wishing to write anything incorrect, I found myself with the following question:

Is the Hopf link a Brunnian link?

According to Wikipedia, a Brunnian link is a link with the property that removing any component produces an unlink (of the appropriate size). That's certainly true of the Hopf link! One could outlaw the Hopf link on size grounds and add "of at least 3 components" to the definition of a Brunnian link, but then the family of rubberband Brunnian links is missing its first member (actually, its second; but the first is a fancy way of drawing the unlink).

This feels a bit like the question "Is 1 prime?" so I suspect that the answer is purely a matter of convention but as I'm not a knot theorist, I don't know what the convention is. So I'm hoping that one Steeped in the Depths of Knot Theory can help me out.

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    $\begingroup$ I feel like the question of whether 1 is prime is more analogous to the question of whether any knot, or the empty manifold, is Brunnian. $\endgroup$
    – S. Carnahan
    Oct 1 '10 at 10:32
  • $\begingroup$ Maybe the correct phrasing should have been to replace "so I suspect" by "in" in the sense that the connection between the questions is very tenuous and only really in the sense that it's a matter of convention (backed up by actual use) rather than any particular deep mathematics. $\endgroup$ Oct 1 '10 at 11:01
  • $\begingroup$ H. Debrunner's "Uber den zerfall von verkettungen" Math. Z. 85 (1964) 154-168 and T. Kanenobu's "Hyperbolic links with Brunnian properties" J. Math. Soc. Japan. Vol 38 No. 2 (1986) 295-308. Are about as close to definitive sources on this as you could ask for. I think perhaps more degenerate would be the debate on whether any knot is considered a Brunnian link. $\endgroup$ Oct 1 '10 at 14:07
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    $\begingroup$ The number 1 is definitely NOT prime. Otherwise, the uniqueness of prime factorizations would fail (and the prime factorization of 1 involves zero primes). $\endgroup$ Oct 1 '10 at 15:18
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    $\begingroup$ Andre, the actual theorem holds whether or not 1 is prime, you just have to rephrase it to "where the factors are primes greater than 1". So it's convention backed up by the fact that more theorems are about "primes other than 1" than "primes together with 1". In some situations, it's irritating that 2 is a prime! "For any odd prime" is quite a common phrase, but not common enough to warrant new terminology. Anyway, that's quite irrelevant to the actual question. $\endgroup$ Oct 1 '10 at 18:08

The Hopf link is normally regarded as Brunnian.

  • $\begingroup$ Thanks! (and some more characters to take me over the limit) $\endgroup$ Oct 1 '10 at 18:04

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