A knot invariant is a useful tool in knot theory. It is a quantity (in a broad sense — some are indeed numbers, some are polynomials, and some are a simple yes/no) defined for each knot. Their usefulness is in distinguishing knots from one another or in outlining other properties of knots. Trefoil knot, the simplest non-trivial knot. ...

Some knot invariants are worked out from a knot diagram, in which case they must be unchanged (that is to say, invariant) under the Reidemeister moves; knot polynomials are examples of this. These are currently the most useful invariants for distinguishing knots from one another, though at the time of writing it is not known whether any of these distinguishes all knots from each other or even just the unknot from all other knots. Trefoil knot, the simplest non-trivial knot. ... This article needs to be cleaned up to conform to a higher standard of quality. ... The unknot, and a knot equivalent to it The unknot is a loop of rope without a knot in it (in knot theory, ropes have no ends; they are loops). ...

Other invariants are defined by choosing a particular diagram, for example, and many take the minimum "value" over all possible diagrams of a knot. This category includes the crossing number, which is the minimum number of crossings for any diagram of the knot. In mathematics, crossing numbers arise in two related contexts: in knot theory and in graph theory. ...

The complement of a knot itself (as a topological space) is known to be a complete invariant of the knot, meaning that it distinguishes the given knot from all other knots up to isotopy. Some invariants associated with the knot complement include the knot group which is just the fundamental group of the complement. In mathematics, the knot complement of a tame knot K is the set-theoretic complement of the interior of the embedding of a solid torus into the 3-sphere. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... An illustration of a homotopy between the two bold paths In topology, two continuous functions from one topological space to another are called homotopic (Greek homeos = identical and topos = place) if one can be continuously deformed into the other, such a deformation being called a homotopy between the two functions. ... In mathematics, a knot is an embedding of a circle into 3-dimensional Euclidean space. ... In mathematics, the fundamental group is one of the basic concepts of algebraic topology. ...

Finally, some invariants are more or less unrelated to diagrams of the knot and need to be worked out in other ways. An example of this is given by the knot genus, i.e. the minimal genus of a Seifert surface spanning the knot. In mathematics, a Seifert surface is a surface whose boundary is a given knot or link. ... In mathematics, a Seifert surface is a surface whose boundary is a given knot or link. ...

New results in recent years about the genus of knots have been obtained from Heegaard Floer homology. This is a homology theory whose Euler characteristic is the Alexander polynomial of the knot. Along a different line of study, there is a combinatorially defined cohomology theory of knots called Khovanov homology whose Euler characteristic is the Jones polynomial. This has recently been shown to be useful in obtaining bounds on slice genus whose earlier proofs required gauge theory. Khovanov and Rozansky have since defined several other related cohomology theories whose Euler characteristics recover other classical invariants. These theories are examples of categorification. In mathematics, homology theory is the axiomatic study of the intuitive geometric idea of homology of cycles on topological spaces. ... In algebraic topology, the Euler characteristic is a topological invariant (in fact, homotopy invariant) defined for a broad class of topological spaces. ... This article needs cleanup. ... Khovanov homology is a Homology theory for knots and links which may be regarded as a categorification of the Jones polynomial. ... This article needs cleanup. ... In mathematics, categorification refers to the process of replacing set-theoretic theorems by category-theoretic analogues. ...