Home > Knot invariant

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.

Some knot invariants are worked out from a 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 the invariants most useful in 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, many take the minimum value over all possible diagrams of a knot. This category includes crossing number, which is the minimum number of crossings for any diagram of the knot. Other invariants which are stronger than the fundamental group are defined by racks and quandles

Finally, some invariants are more or less unrelated to diagrams of the knot and need to be worked out in other ways. For example, the genus of a knot.