Knot invariants

In the mathematical field of knot theory, a knot invariant is a quantity (in a broad sense) defined for each knot which is the same for equivalent knots. The equivalence is often given by ambient isotopy but can be given by homeomorphism. Some invariants are indeed numbers, but invariants can range from the simple, such as a yes/no answer, to those as complex as a homology theory . Research on invariants is not only motivated by the basic problem of distinguishing one knot from another but also to understand fundamental properties of knots and their relations to other branches of mathematics.

From the modern perspective, it is natural to define a knot invariant from a knot diagram. Of course, it must be unchanged (that is to say, invariant) under the Reidemeister moves. Tricolorability is a particularly simple example. Other examples are knot polynomials, such as the Jones polynomial, which are currently among the most useful invariants for distinguishing knots from one another, though currently it is not known whether there exists a knot polynomial which distinguishes all knots from each other. However, there are invariants which distinguish the unknot from all other knots, such as Khovanov homology and knot Floer homology.

Other invariants can be defined by considering some integer-valued function of knot diagrams and taking its minimum value over all possible diagrams of a given knot. This category includes the crossing number, which is the minimum number of crossings for any diagram of the knot, and the bridge number, which is the minimum number of bridges for any diagram of the knot.

Historically, many of the early knot invariants are not defined by first selecting a diagram but defined intrinsically, which can make computing some of these invariants a challenge. For example, knot genus is particularly tricky to compute, but can be effective (for instance, in distinguishing mutants).

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