Crossing number

Remarks

Since the minimum is taken over all knots equivalent to $K$, the number $c(K)$ is a knot invariant. Example: $c(\text{unknot}) = 0$, since the round circle has a diagram with no crossings. For the trefoil, $c = 3$: it has a diagram with $3$ crossings, and every diagram with at most $2$ crossings represents the unknot, which is not equivalent to the trefoil. Non-example: a diagram of the unknot with one twist has $1$ crossing, but $1$ is not the crossing number of the unknot, because the minimum is $0$. For a reduced alternating diagram, the number of crossings equals the crossing number; this was one of the Tait conjectures, proved with the Jones polynomial. The same definition applies to links.