Knot Theory
Knot theory is a branch of topology that studies the mathematical properties of knots. A knot is a closed curve that is embedded in three-dimensional space. The study of knots has a rich history, dating back to the 19th century, and has many applications in fields such as physics, chemistry, and biology.
Definitions
A knot is a closed curve that is embedded in three-dimensional space. In other words, it is a curve that does not intersect itself and is not allowed to pass through any object in space. A knot is considered to be the same if one can be transformed into the other by a continuous deformation, without cutting or passing through itself.
A link is a collection of knots that are linked together. For example, a chain is a link, and each of its links is a knot.
A projection of a knot is a two-dimensional representation of the knot on a plane. A projection can be obtained by projecting the knot onto a plane and then drawing it as a curve on the plane, with crossings indicated by overpasses or underpasses.
Knot Invariants
A knot invariant is a property of a knot that is preserved under the homeomorphism. In other words, if two knots are homeomorphic, then they have the same knot invariant.
One of the most important knot invariants is the knot polynomial, which assigns a polynomial to each knot. The simplest example of a knot polynomial is the Alexander polynomial, which is a polynomial in one variable that can be computed from the knot's projection. The Alexander polynomial is invariant under the homeomorphism.
Another important knot invariant is the Jones polynomial, which is a more complicated polynomial that can be computed from the knot's projection. The Jones polynomial is also invariant under the homeomorphism.
Applications
Knot theory has many applications in different fields. In physics, knots are used to model the behavior of polymers and DNA molecules. In chemistry, knots are used to study the behavior of molecules and chemical reactions. In biology, knots are used to study the structure of proteins and the folding of RNA molecules.
Knot theory is also important in topology and geometry. It provides a way to classify and study three-dimensional manifolds, which are spaces that are locally like three-dimensional space.
Conclusion
Knot theory is a fascinating branch of topology that studies the mathematical properties of knots. It has many applications in different fields, and has contributed to our understanding of the structure and behavior of molecules, proteins, and DNA. Knot invariants such as the Alexander polynomial and Jones polynomial provide a way to distinguish between knots that are homeomorphic, and provide a powerful tool for studying the properties of knots.