Where Do We Get Our Feet Wet?

Professor Korner has generously supported school mathematics for more than 30 years and has been a good friend to NRICH since it started.

Links and Knots

Some puzzles requiring no knowledge of knot theory, just a careful inspection of the patterns. A glimpse of the classification of knots, prime knots, crossing numbers and knot arithmetic.

Earth Shapes

What if the Earth's shape was a cube or a cone or a pyramid or a saddle ... See some curious worlds here.

Exhibition of Knots

Stage: 5

Published December 1998,March 1998,December 2011,February 2011.

A visit to this site opens a door to a new world and beautifully illustrates how mathematics may be used to explore and understand the seemingly commonplace knot. The site is clearly set out as an exhibition, presented by the School of Mathematics of the University of Wales, Bangor, and designed by Professor R.Brown, Nick Gilbert and Tim Porter. If you follow the pages or sections under the Contents title, the site will carefully take you through the introduction, definitions, understanding and application of knots.

The section names are intriguing and include 'Sorting out the bowline' , 'The Unknotting Number' and 'Colouring Knots' ; however, I would strongly suggest that unless you are familiar with the mathematics of knots, you should follow the order set out in the Contents. The presentation is very clear and you do not have to be mathematically trained to appreciate the knots. To help your understanding of various terms used in mathematics, an icon at the bottom of the page of each section will take you to a linked page that defines the terms being used in the sections; e.g. Analogy. In the 'Knots and Numbers' part of the contents an analogy is made between adding knots and the multiplication of numbers. Again, in the section 'Prime Knots' a further analogy is made between prime numbers and knots. Not to be missed are the Applications Sections, where knots are related to such areas as fluid flows, String Theory (take note all cosmologists and physicists) and to DNA (a rather surprising link for me), with good links to other sites for those interested in the applications. There are further links to yet more sites related to knots on Nick Gilbert's home page, which is accessible from this site.

I have but one slight criticism. In the section titled 'Mirror Images' a rather sudden jump to notation of the image and its invariants is made without defining what the terms meant in $l^2 m^2-2l^2-l^4$. This did rather stop my education in knots for the moment. However, links with other sites did allow me to continue to peruse knots to a greater depth. I would have liked a little more about the theory of colouring than was provided by the site, but here again to quote the designers; 'We are not interested in conveying technique '. This would seem to agree with their stated aim of 'suggesting that the making of mathematics is a natural human activity, part and parcel of the usual methods by which man has explored, discovered and understood the world'. In this last aim, I think the designers have most definitely succeeded.

In conclusion, this site is an important addition to the world population of sites if only for the successful presentation of 'how mathematics gets into knots' or how mathematics is done, and I'm sure that it will have you reaching for your colour pencils.