linkinvariants

Link invariants are mathematical quantities associated with a link in three-dimensional space that remain unchanged under ambient isotopy. An ambient isotopy is a continuous deformation of the surrounding space that does not self-intersect, and thus preserves the topological structure of the link. This means that if two links can be smoothly deformed into one another without cutting or passing through themselves, they will have the same link invariants.

The primary purpose of link invariants is to distinguish between different links. If two links have different values for a particular invariant, then they are guaranteed to be topologically distinct. However, if they share the same value for a set of known invariants, it does not necessarily mean they are isotopic; the set of invariants may not be complete.

Several types of link invariants exist, ranging from simple scalar values to more complex algebraic structures. Examples include the linking number, which for a two-component link measures how many times one component winds around the other, and the Alexander polynomial, a polynomial in a single variable that encodes topological information about the link. More sophisticated invariants, such as knot polynomials (e.g., Jones polynomial, HOMFLY polynomial) and homology groups (e.g., Khovanov homology), provide deeper insights into the structure of links. These invariants are crucial tools in knot theory and have found applications in fields like physics and chemistry.