# Dmitri Kozlov

Knots have been the subject of traditional art since ancient time.
Woven patterns, stone and wood carving and other 2D knotted ornaments
were very popular throughout the world. Today artists depict knots
mainly as 3D objects to express the interaction of knots with the
surrounding space.

My approach is based on the idea that cyclic periodic knots made of
resilient filaments like steel wire behave as kinetic structures.
Knots tied with such materials must have a large number of physically
contacting crossings namely the vertices of surfaces. The crossings
slide along the resilient filaments which at the same time twist
around their central axis. The waves on the filaments move and change
their lengths to adapt to the current disposition of the contact
crossings. Thanks to these properties the knots change their geometry
as a whole and create vertex or point surfaces with an arbitrary
Gaussian curvature. I designated as NODUS-structures the complicated
knots of this type.

This transformable spherical NODUS structure is based on the
principle of a complicated periodic knot known as chain
Turk’s-Head. It has 17 loops and two 17-gons at the polar
openings. I chose the number 17 because it symbolizes the early
discovery of Karl Friedrich Gauss that states 17-gon can be
constructed with a ruler and compasses.

The name of Gauss is closely connected with the beginning of knot
theory and the theory of surfaces. As opposed to solid surfaces
that can not change their curvatures without breaks and folds, the
point surfaces of NODUS-structures can be changed from the
positive Gaussian curvature (elliptic) to the negative one
(hyperbolic) through the neutral (parabolic) curvature and vice
versa due to the torus rotation. NODUS-structures, as outer
(elliptic) or an inner (hyperbolic) parts of torus, can rotate
around the imaginary circle axis. The structure can turn inside
out and take the forms of the outer and the inner parts of the
torus surface.

Any physical knotted filament has a thickness, so any real knot is
a torus. A non-knotted torus or simple ring is known as a trivial
knot. There is a special class of torus knots that can be placed
on the surface of a torus without any self-crossings. The simplest
trefoil is an example of torus knots.

A trefoil knot may have two mirror types: a left and a right. Each
of them can be tied on the torus surface without self crossings,
but when tied together on the same torus, they cross each other
and form an elementary knotted fabric on the torus surface. If
both knots are made of resilient filaments and their crossings are
really contacting, they form the structure of elementary toroidal
self-supporting point surface.

My work describes the principle of materialization of two mirror
torus knots. I took two mirror torus knots with parameters q=17
and p=16 and wove them into a self-supporting kinetic toroidal
NODUS-structure. The structure may take planar and
three-dimensional forms.

Knots are closely connected with one-side surfaces. The edge of a
Möebius band made of a ribbon twisted through π angle is a
continuous curve namely a trivial knot. Then the same ribbon is
twisted through 3π angle its edge is the trefoil knot.

Another approach to the shape of Möebius band arise from the
cutting of Klein bottle on two equal parts. These parts have the
form of Möebius band with self-crossings.

In turn the Klein bottle can be done as a torus with figure eight
cross section twisted through π angle and thereby can be presented
by two mirror torus knots as explained in my previous
description.

These two mirror torus knots on the imaginary Klein bottle surface
can be cut along the line of the equator and divided into two
cyclic knots or links with self-crossings.

The same idea is reflected in this my work. A cyclic knot made of
a single piece of steel wire forms a NODUS-structure with
self-crossing and one edge. It represents a Möebius band as a half
Klein bottle.