Knotenlehre

Knotenlehre is the branch of mathematics that studies the properties and behavior of knots and links. The term "Knotenlehre" is German for "theory of knots". The subject has its roots in topology, a field of mathematics that deals with the properties of shapes and spaces that are preserved under continuous deformations.

Knotenlehre emerged as a distinct area of study in the 19th century, particularly through the work of mathematicians such as Augustin-Louis Cauchy and Peter Guthrie Tait. These early researchers developed the foundation for the understanding of knots as abstract objects, distinct from their physical manifestations. This allowed for the consideration of knots as mathematical entities, governed by their intrinsic properties rather than their physical properties.

One of the central concerns in Knotenlehre is the classification of knots and links, which involves analyzing their invariants. These invariants can include, for instance, the number of crossings of the knot diagram, the Alexander polynomial, and the Jones polynomial. The study of invariants has led to significant discoveries, such as the understanding of knot categories, the theory of braid groups, and the concept of knot boundary.

The mathematical treatment of Knotenlehre draws on various areas of mathematics, including geometry, algebra, and combinatorics. Its application can be seen in various fields, including materials science, biology, and physics. Today, Knotenlehre is a relatively isolated but well-established area of mathematics, with its fundamental concepts widely applied across branches of mathematics and the natural sciences.