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Knot InvariantA 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. 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. 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. 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. 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. 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 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. These theories are examples of categorification.
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