Crossing number

In knot theory, the crossing number is an example of a knot invariant. A knot's crossing number is simply the lowest number of crossings of any diagram of the knot.

By way of example, the unknot has crossing number zero, the trefoil knot three and the figure eight knot four. There are no other knots with a crossing number this low and just two knots have crossing number 5, but the number of knots with a particular crossing number increases rapidly as we go higher.

Knots (to be precise prime knots are traditionally indexed by crossing number, with a subscript to indicate which particular knot out of those with this many crossings is meant (this sub-ordering is not based on anything in particular). The listing goes 3₁ (the trefoil knot), 4₁ (the figure-eight knot), 5₁, 5₂, 6₁, etc. This order has not changed significantly since P. G. Tait published a tabulation of knots in 1877.