Physical knot theory
Encyclopedia
Physical knot theory is the study of mathematical model
s of knotting phenomena, often motivated by physical considerations from biology
, chemistry
, and physics
. Traditional knot theory
models a knot as a simple closed loop in three dimensional space. Such a knot has no thickness or physical properties such as tension or friction
. Physical knot theory incorporates more realistic models. The traditional model is also studied but with an eye toward properties of specific embeddings ("conformations") of the circle. Such properties include ropelength
and various knot energies
.
Mathematical model
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used not only in the natural sciences and engineering disciplines A mathematical model is a...
s of knotting phenomena, often motivated by physical considerations from biology
Biology
Biology is a natural science concerned with the study of life and living organisms, including their structure, function, growth, origin, evolution, distribution, and taxonomy. Biology is a vast subject containing many subdivisions, topics, and disciplines...
, chemistry
Chemistry
Chemistry is the science of matter, especially its chemical reactions, but also its composition, structure and properties. Chemistry is concerned with atoms and their interactions with other atoms, and particularly with the properties of chemical bonds....
, and physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
. Traditional knot theory
Knot theory
In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In precise mathematical language, a knot is an embedding of a...
models a knot as a simple closed loop in three dimensional space. Such a knot has no thickness or physical properties such as tension or friction
Friction
Friction is the force resisting the relative motion of solid surfaces, fluid layers, and/or material elements sliding against each other. There are several types of friction:...
. Physical knot theory incorporates more realistic models. The traditional model is also studied but with an eye toward properties of specific embeddings ("conformations") of the circle. Such properties include ropelength
Ropelength
In knot theory each realization of a link or knot has an associated ropelength. Intuitively this is the minimal length of an ideally flexible rope that is needed to tie a given link, or knot...
and various knot energies
Knot energy
In physical knot theory, a knot energy is a functional on the space of all knot conformations. A conformation of a knot is a particular embedding of a circle into three-dimensional space. Depending on the needs of the energy function, the space of conformations is restricted to a sufficiently...
.