nLab crossing number

Definition

Let $K$ be a knot.

Definition

The crossing number, $c(K)$, of $K$ is the minimum number of crossings in a knot diagram in the isotopy class of $K$.

The crossing number is thus the number of crossings in the simplest picture of a knot. A diagram of a knot $K$ with exactly $c(K)$ crossings is called a minimal diagram.

Examples

• for the unknot: $c(unknot) = 0$;

• for the trefoil knot: $c(trefoil) = 3$;

• for the figure eight knot: $c(figure-8) = 4$.

• if a knot diagram has $1$ or $2$ crossings it represents the unknot, so there are no non-trivial knots with $c(K) = 1$ or $2$.

The crossing number is related to the unknotting number, but in quite a subtle way.

Related concepts

• writhe

• linking number

• framing number

References

In the books by Burde and Zeischang (1985) and Kauffman (1987), the tables of knots are arranged according to crossing number. (Choices have been made of one mirror image or the other.) Given some arbitrary diagram, the crossing number of the knot that it respresents may be hard to determine.