Forces and momenta of forces within
the most tight trefoil knot
Sylwester Przybyl
Poznan University of Technology, Poland
The Finite Element Method has been used to find the most tight
conformation of the trefoil knot tied on the ideal rope. The processed
knot contains N = 100320 vertices. Each vertex is connected via
longitudinally elastic beams with other vertices. The forces with
which the beams act on the vertices shift them slowly in such a way
that in the final conformation all forces acting on each of the vertices
sum up to zero.
There are three types of the beams:
1.
axis beams connecting given vertex with the neighbor vertices;
2.
contact beams that prevent the rope from overlapping;
3.
curvature beams that limit the curvature to 1/R.
Forces acting along the axis beams can be seen as tension. At
each vertex we have two such forces. The beams have identical
elastic properties. Their natural lengths are different, but when
subject to tension they elongate to reach finally the same length. For
N = 100320 the final length of the axis beams dl = 0.000326.
Fig. 4. Normal forces stemming from the axial tension (red)
and the sum of the reaction forces exerted by the
contact beams (blue).
The contact beams have lengths DC  2R, where R is the
radius of the rope. The beams are compressed by forces stemming
from the tension present within the axis beams what makes them
eventually reach the identical length equal 2R.
Beam of type 3 are of various lengths.
The figures bellow present the beams and the forces.
Fig. 5. The sum of the normal forces stemming from
the axial tension and the reaction forces stemming
from the compression of the contact beams.
Fig. 1. The beam skeleton of the knot and the tension
forces acting along the axis breams.
Fig. 6. Curvature controlling
beam forces.
Fig. 7. Construction of the
curvature controlling beams.
REFERENCES
Fig. 2. Normal forces
stemming from tension.
Fig. 3. The reaction forces (blue) exerted
on vertices by the contact beams.
Mathias Carlen, Computation and Visualization of Ideal Knot
Shapes, PhD thesis, Lausanne 2010.