Linkinvarianten

Linkinvarianten (plural for "linkinvariant") are fundamental concepts in knot theory, a branch of topology that studies mathematical knots. A linkinvariant is a quantity or mathematical object assigned to a link—a collection of one or more knots—such that it remains unchanged under ambient isotopies. Ambient isotopies are continuous deformations of the link in three-dimensional space that do not involve cutting or passing strands through each other.

The purpose of linkinvariants is to distinguish between different links or knots. If two links have different values for a particular linkinvariant, they are not equivalent. Conversely, if their invariants coincide, the links are potential candidates for being equivalent, although additional invariants may be necessary for a definitive classification.

Common examples of linkinvariants include the linking number, the Jones polynomial, the Alexander polynomial, and the HOMFLY polynomial. The linking number measures how many times two components of a link wind around each other. Polynomial invariants, such as the Jones and Alexander polynomials, encode complex topological information in algebraic form, allowing for more nuanced distinctions between links.

Calculating and analyzing linkinvariants plays a critical role in understanding the topology of knots and links, with applications ranging from molecular chemistry to the study of DNA strands. These invariants provide a powerful toolbox for mathematicians to classify and detect properties in the study of three-dimensional spaces.

Overall, linkinvarianten serve as essential tools in knot theory, aiding in the classification and analysis of knots and links in mathematical and applied contexts.