The trefoil knot can be laid out so that it has three crossings. The knot may be manipulated to increase the number of crossings, but no amount of
manipulation will decrease the number of crossings. The trefoil knot therefore has crossing number three. However, the figure of eight knot has crossing number four. This shows that the trefoil knot and the figure
of eight knot must be different knots.
Crossing number alone is not sufficient to distinguish between all prime knots.
Although the trefoil knot is the only knot with crossing number three and the figure of eight knot is the only knot with crossing number four, there are
two different prime knots with crossing number five and three with crossing number six, as shown in the picture on the left, so to distinguish between these knots mathematically requires more sophisticated methods.
The first attempt at knot tabulation was made by the Reverend Thomas P. Kirkman in the 1880s. While still at school the English mathematician John Conway
became interested in the mathematical properties of knots. In 1969, Conway adapted his earlier ideas to invent a new knot notation which he used to determine all the prime knots with 11 or fewer crossings.
Another English mathematician, Morwen Thistlethwaite developed a computer program that by 1982 had tabulated all the prime knots with up to 13 crossings.
The number of prime knots with each crossing number is as follows:
3 4 5 6 7 8 9 10 11 12 13
1 1 2 3 7 21 49 165 552 2176 9988
Immortality and Gordian Knot animations by John Robinson and Nick Mee. Text and other animations by Nick Mee from the CDROM 'Art and
Mathematics'.
