Knot invariant
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In the mathematical field of knot theory, a knot invariant is a quantity (in a broad sense) defined for each knot. Some invariants are indeed numbers, but invariants can be as simple as a yes/no answer or as complicated as a homology theory . Research on invariants has been motivated by a desire to find means to distinguish one knot from another, but often also to understand fundamental properties of knots and their relations to diverse branches of mathematics.
Some knot invariants are defined according to a given 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 among 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.
Other "diagrammatic" 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.
A number of knot invariants are not defined diagrammatically, although for very special classes of knots (such as alternating knots) one can sometimes compute such an invariant directly from the diagram. An example of this is given by the knot genus, i.e. the minimal genus of a Seifert surface spanning the knot.
The complement of a knot itself (as a topological space) is known to be a complete invariant of the knot by the Gordon-Luecke theorem, meaning that it distinguishes the given knot from all other knots up to isotopy and reorientation of the ambient 3-space (thus, for example, the two trefoil knots are mirror images of each other, so their complements are homeomorphic, but they are not isotopic). Some invariants associated with the knot complement include the knot group which is just the fundamental group of the complement. Most knots are hyperbolic, which means the hyperbolic volume is an invariant for these knots. Volume, and other hyperbolic invariants, have proven very effective, utilized in some of the extensive efforts at knot tabulation.
In recent years, there has been much interest in homological invariants of knots. Heegaard Floer homology is a homology theory whose Euler characteristic is the Alexander polynomial of the knot. It has been proven effective in deducing new results about the classical invariants. 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.
The Fary-Milnor theorem states that if the total curvature of a knot K in 
satisfies
where κ(p) is the curvature at p, then K is an unknot. Therefore, for knotted curves,
