SYMMETRY BREAKING IN TORUS KNOTS
SMALL TORUS KNOTS
Let us consider
one of the simplest type of prime knots: torus T2,n knots, whose
n-fold symmetrical conformations are easily defined parametrically. For n=3,
i.e. the 31 (trefoil) knot, its final, tight conformation preserves
the 3-fold symmetry of the initial conformation. For n=5, i.e. the 51
knot, the 5-fold symmetry of the initial conformation becomes broken during
the tightening process. See figure below.
LARGE TORUS KNOTS
For much larger n, the evolution of
the symmetrical, parametrically defined torus knots towards their most tight
conformations is a complex process. Figures below show its consecutive stages
in the cases of n=33 and n=99.
n=33
n=99
Further simulations performed
with larger T2,n torus knots prove that the shapeless, blob-like
conformation shown at the last frame above is not the best one. Starting from
a different initial conformation of the knot we find a different final conformation,
which proves to be slightly better from the point of view of the length of
the rope used to tie it. See below.
Initial linear conformation
Twisted final conformation
OPEN QUESTION: Are there any better
conformations?
The problem of ideal T2,n
torus knots is closely related to the problem of the ideal twisted pair.
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