Kim Whittlesey, a fellow grad student at Berkeley, needed to solve this problem for her Ph.D. dissertation. The answer is yes, they exist for arbitrary numbers of letters: just take commutators. Define [a,b] = a b a’ b’. This is nontrivial in the free group, but becomes trivial if either a or b is. Then [[[[[a,b],c], d], e], f], for example, is a nontrivial word in the free group, but becomes trivial if you mod out by any of a/b/c/d/e/f. On Fri, Oct 26, 2012 at 10:39 AM, Adam P. Goucher <apgoucher@gmx.com> wrote:
Here's an interesting group-theoretic puzzle with connections to Borromean rings, Brunnian braids et cetera:
http://cp4space.wordpress.com/2012/10/26/borromean-strings/
I doubt the existence of Borromean strings, although I haven't been able to prove it.
Sincerely,
Adam P. Goucher
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