nLab
crossing number

Let KK be a knot.

Definition

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

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

Examples
  • c(unknot)=0c(unknot) = 0;

  • c(trefoil)=3c(trefoil) = 3;

  • c(figure8)=4c(figure-8) = 4.

  • if a diagram has 11 or 22 crossings it represents the unknot, so there are no knots with c(K)=1c(K) = 1 or 22.

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

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.

Created on October 15, 2010 17:07:09 by Tim Porter (95.147.236.87)