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Knots have a long history in art. Intricate knot designs illuminate beautiful Celtic manuscripts such as the Book of Kells created by Irish monks well over 1000 years ago. Many great Renaissance artists including Leonardo, Durer and Raphael also constructed elaborate knot designs. Several of the works of the contemporary sculptor John Robinson are based on knots. The animation above shows John Robinson's sculpture 'Immortality' which has the structure of a trefoil knot twisted into a Mobius strip. The animation on the right shows John Robinson's 'Gordian Knot'.

Lord Kelvin proposed that the fundamental constituents of matter, might be vortex loops in the aether, similar to smoke rings. To explain differing chemical properties, Kelvin suggested that the vortex loops were knotted in various ways, so that a hydrogen atom might have the structure of a trefoil knot, for instance, while an oxygen atom might have the structure of a figure of eight knot.

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Of course, it has turned out that Kelvin's theory is completely wrong, but this was not obvious at the time. In order to investigate the idea further, mathematicians such as Peter Tait set out to find methods to distinguish different knots mathematically. Solving this problem has turned out to be far more difficult than was initially supposed.

When mathematicians consider a knot they think of it as a closed loop, with the two ends joined so that the knot cannot be untied without cutting the loop.

Two knots are equivalent if they can be transformed into each other by any manipulations that do not involve cutting the knots. The simplest of all knots is the overhand knot or trefoil knot which crosses itself three times. The animation above shows a trefoil knot.

If a knot can be cut in two places, so that the four loose ends can be connected together to produce two other knots, then the original knot is a compound knot. For instance, if we follow this procedure with a granny knot we can produce two trefoil knots. Granny knots are therefore compound knots. The prime knots are those that cannot be decomposed in this way into other knots. In the same way that all whole numbers can be factorised into products of prime numbers, so all knots are constructed from the prime knots.

To distinguish different knots it is necessary to find mathematical properties that differ from knot to knot. The simplest such property is crossing number. If a knot is laid out flat, the crossing number is the minimum number of times that the knot crosses over itself for all possible arrangements of the knot.

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 CD-ROM 'Art and Mathematics'.