Elastic bands

[Laurent Bartholdi]

I've been following the discussion on the rubber-bands link problem, and
wish to react: it seems to me that part of the difficulty is realizing that
it is a hard problem, for which mathematics is necessary. Witness the
complicated unknots in
http://homepages.math.uic.edu/~kauffman/IntellUnKnot.pdf : certainly most
of us would believe these cannot be unknotted, unless we knew we were being
tricked.

There are two steps: one is the conversion of the 3d figure into something
discrete; for example, a presentation of the fundamental group of the
complement. The second step is finding an appropriate invariant of the
group. For example, it seems (experimentally) that the class-2 nilpotent
quotient always has 2-torsion, while for a trivial link it would be
torsion-free. It's easy to prove in individual cases, though a single
argument working for all n is a bit more tricky.

The first step may raise some FOM issues, since it essentially relies on
Reidemeister / Wirtinger connections between topology and algebra. The
second step is just classical mathematics. I don't think any artificial
intelligence can help there: it would too easily be tricked by the unknots,
in Kauffman-Lampropoulou's article or Haken's "Gordian knot".
-- 
Laurent Bartholdi     laurent.bartholdi<at>gmail<dot>com
Mathematisches Institut, Georg-August Universität zu Göttingen
Bunsenstrasse 3-5, D-37073 Göttingen, Germany
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