# Classification

Making lists (such as the table of knots) is a basic human activity. However, in view of the complexities of the world, you cannot make a list of everything, at least not if you are to lead a sensible life.

You may recall the Memory Man described by Luria in [Luria, A.R., (1968), The mind of a mnemonist, trans. by Lynn Solotaroff, Basic Books, New York.] who was unable to forget anything, and consequently was not able to lead a normal life. One of the diagnostic features for autistic children is that they remember nonsense patterns as easily as other patterns [Wing, Lorna, (1976), Early childhood autism; clinical, educational, and social aspects, Pergamon Press, Oxford].

So in making lists we impose or find order: we classify. For example, a zoologist does not list all the animals in a game reserve, he lists antelopes, elephants, lions and so on. In order to do so, he needs criteria for saying that two animals are the same. In mathematics the notion of equality, or, in more precise mathematical terms, equivalence, is basic.

Knot theory presents interesting mathematical points in this area. Firstly, "When are two knots the same?" is a non trivial question. Secondly, to make an initial list of the first elements of an infinite family involves some classification into the "simplest" elements. Thirdly, the presentation of even such a simple list is likely to suggest the need for some further order and classification.

We address the first question by showing in our boards how diagrams of knots can be transformed, without changing the knot.

This is one area where computer animated graphics could have greatly improved the presentation. However, we were limited by the cost of producing the graphics, and the cost of presenting the graphics at an exhibition display. The latter is not so hard to overcome, since a video could be made and easily displayed, for example at most schools. It is more important for us to show the context in which any computer graphics would run, as this would dictate what graphics should be produced in order to support the overall themes.

One of the goals that we set ourselves was to explain the meaning of a list of knots. Although such a list is apparently simple, the explanation involves the following ideas:

• when are two knots the same;
• crossing number;
• mirror images;
• the arithmetic of knots;
• prime knots.

These themes give interconnections between the different boards.

We have discussed when two knots are the same, and Reidemeister moves in Same Knots. To decide when two knots are not the same needs the notion of Invariant

© Mathematics and Knots, U.C.N.W.,Bangor, 1996 - 2002
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