**Symmetry of the ideal knots**

In contradiction to our expectations,
**ideal
knots as a rule** **do not have any symmetry elements**. It seems,
however, that there are a few exceptions. I wrote "it seems" on purpose.
As a matter of fact, we still know very little about ideal knots. Numerical
simulations provide us with some apparently most tight conformations. We
calculate their parameters such as the average crossing number or writhe.
But that is all. In a few cases we feel justified to believe that the conformation
we have found is very close to the Ideal Conformation. Basing on this belief
we may try to guess some properties of the latter, for instance its symmetry
point group. And this is what I shall try to achieve below.

The ideal conformation is know
for sure just but in one case - the unknot. **The ideal unknot is a perfect
circle**. Thus, symmetry properties of the ideal unknot are known.

Let us consider the 3_{1}
(trefoil) knot. Tightening it with the SONO algorithm leads inevitably
to a conformation, see figure below, the symmetry point group of which
seems to be D_{3}. (How to prove it?)

What about the 4_{1}
(figure eight) knot? Also here SONO algorithm ends with conformation
of always the same kind, see figure above. As noticed by Vsevolod Katritch
the ideal 4_{1} knot has most probably the S_{4} symmetry
point group . Let me demonstrate it.

Take the most tight conformation
of the 4_{1 }knot found by the SONO algorithm, rotate it by 90
degrees and reflect it in a mirror plane perpendicular to the rotation
axis. You will find that the result coincides with the initial conformation.
Figure below clearly presents this.

(Ends of the rope on which
the knot is tied were not closed on purpose - the defect allows one to
see better symmetry operations.)