Invariants

Classification of knots involves two aspects:

• when are two knots the same?
• and when are they not the same?

The first usually involves transforming one diagram of the knot into another. The second involves the more subtle point of deciding when such a transformation is not possible. Such a decision involves the notion of invariants.

We deal with four invariants in our presentation:

• crossing number;
• unknotting or Gordian number;
• bridge number;
• colouring number.

We also mention briefly the new knot polynomials which enable one to distinguish easily between a trefoil and its mirror image. The advantage of the four invariants we deal with in detail is that they can be easily presented at this level, and that they suggest many detailed exercises and examples which people can try for themselves.

The further point made by the discussion of invariants is that we do not claim to give a complete set of invariants, that is, we do not have some method of distinguishing all possible knots. A totally different approach to invariants is given by the method of paths and loops. Again, this does not give a complete set of invariants.

Thus many problems remain in the theory, and this again is a point which is easily conveyed. We do want the reader to see that mathematics is, and will continue to be, an open-ended activity.

More generally, mathematicians are not among those who expect that some new theory will come along which will somehow answer all our questions, a kind of `general unified problem solver'. We do expect to find new ways of looking at and solving old questions, to find new questions, and to find new, surprising and beautiful intricacies of patterns, structures and relationships at which to marvel.