Knotenvector

Knotenvector is a mathematical concept used in the study of knots and links in topology, a branch of mathematics concerned with the properties of space that are preserved under continuous deformations. The term "Knotenvector" refers to a specific type of vector space associated with a knot or link, which encodes certain topological and geometric properties of the knot.

The Knotenvector is constructed using the Jones polynomial, a powerful invariant in knot theory that assigns a polynomial to each knot or link. The Jones polynomial can be expressed as a Laurent polynomial in a variable t, and the coefficients of this polynomial form the components of the Knotenvector. This vector space is finite-dimensional and its dimension is related to the complexity of the knot.

One of the key applications of the Knotenvector is in the classification of knots and links. Since the Knotenvector is an invariant, two knots with different Knotenvectors are necessarily distinct. Furthermore, the Knotenvector can be used to distinguish between knots that have the same Jones polynomial but differ in other invariants.

The Knotenvector also plays a role in the study of knot concordance, a notion that generalizes the concept of knot equivalence. Two knots are concordant if one can be transformed into the other through a sequence of ambient isotopies and additions or removals of unknotted components. The Knotenvector can be used to detect non-concordance between knots.

In summary, the Knotenvector is a valuable tool in knot theory, providing a means to encode and compare the topological properties of knots and links. Its construction and applications are closely tied to the Jones polynomial and other invariants in the field.