(ABSTRACT of a talk at the IMACS World Congress)
Piotr Pieranski, S. Przybyl and Andrzej Stasiak
Session nr.: 152
Paper nr.: 11
What, from the point of view of the knot theory,
was the Gordian Knot? Was it a closed or open knot?
As the legend says, nobody was able to untie it. Untie? As we know, any open knot can be untied – the algorithm to untie an open knot is known, one may lack patience, not skill, to untie any knot. On the other hand, had the Gordian Knot been a closed knot, no one would have chance to untie it, no matter what type of a knot it had been.
As we know well, even the trefoil knot cannot be transformed into the trivial knot without cutting the rope. It could not be so simple. Most probably the problem of the Gordian Knot was totally different.
As one of us (A.S.) suggested, it could have been the problem not of untying but disentangling a knot.
The simplest problem one can pose here is: are there such a entangled conformations of the unknot tied on a finite piece of rope, which cannot be disentangled to the ground state torus conformation?
This is a non-trivial problem. A firm answer to the question is not easy to find. As the numerical simulations we performed indicate, the answer is positive: there exist conformations of a the unknot, entangled in such a manner, that they cannot be disentangled to the torus conformation. We present an interesting, most probably simplest example.
Have a look at the Kusner's and Sullivan's "Moebius-invariant knot energies". (Chapter 17 of The ideal knots, eds. A. Stasiak, V. Katritch and L.H. Kauffman, World Scientific, Singapore 1998.) At Figure 6 you will find there a picture of an entangled conformation of the unknot. It was proposed by Freedman, He and Wang. The conformation looks nasty, but as Kusner and Sullivan demonstrated, it can be disentangled by a program looking for the minimum of the Moebius energy of the knot.
"Is SONO able to disentangle it?" - asked Andrzej Stasiak. "I bet a bottle of wine, NOT!" - he added.
To find out who won the bottle, have a look at
this short movie (1.2 Mb).
If you want to see its larger, better quality version click here. But be patient - the GIF file is rather large: 6 Mb.
As we have demonstrated, a slightly more complex version of the Freedman's conformation leads to a tight conformation which proves to be Gordian.
SONO is not able to disentangle it.
We think, nobody is able to disentangle it without cutting the rope.
A formal proof does not exist.
Nauki (in Polish)
Pour la Science
Tony Phillips, The Gordian Unknot, American
Keith Devlin, Untying the Gordian Knot, The Mathematical Association of America
Keith Devlin, Unravelling the myth, The Guardian
Mike Martin, Legendary knot finally untied, UPI Science News