In mathematics, knots are usually described by a continuous injective map from the circle into three dimensional Euclidean space; we sometimes require these maps are better behaved, for example piecewise linear, to avoid particularly pathological cases. This corresponds to what we would usually call knots, except that we require both ends to be attached so that there is no way of untying it without breaking it or passing it through itself.

Once this is defined, we need to get an idea of how to compare different knots - there are many ways of doing this and saying which knots are equivalent, for example we can use a homeomorphism of onto ; or we can alternatively use the Reidemeister moves on the knot diagrams.

One of the main problems in the study of knots was that of classifying knots, often by the minimum number of crossings one would need to have in any diagram of a knot (a well behaved projection of the knot onto a plane with under and over crossings). The hardest thing in making up these lists often comes down to figuring out if two different diagrams give the same knot - a nice way of telling the difference is by computing particular invariants attached to the knot - it might be a group (the knot group), a polynomial (the Alexander polynomial, Jones polynomial and many others), a number (genus, crossing number, ...).

A related area is the study of links - a link is like a knot except it might be constructed out of more than one circle. A basic invariant in this case is the linking number, telling us how much the different components are linked together.

Join us in MS.04 at 19:30 for an overview of knot theory, including descriptions of all the knot invariants described above. Then we untie ourselves and go to the pub!

(PS. The next discussion group is going to be David Mond's talk on Wednesday 13th, then no discussion group on Thursday 14th)