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STMC 2009 Poster Round


Wall decoration in Ste Trinité in Paris Wall decoration in Ste Trinité in Paris

The two pictures above show parts of the interior wall decorations in the church of Ste Trinité in Paris. The central portion of each image may be redrawn as shown below.

Link diagram for left-hand picture Link diagram for right-hand picture

We consider these to be diagrams of links. Each diagram shows some loops (say, of string) which are interlinked.

The Borromean rings

The Borromean rings

The left-hand diagram corresponds to the three linked loops shown alongside.

This link is often referred to as the Borromean rings. All three of the loops are definitely linked: there is no way to separate them without cutting.

Question 1
What are the essential differences between the two links shown in the church pictures?

Interchanging crossings

There is a simple way to change the left-hand diagram above to obtain the right-hand diagram: interchange crossings.

To interchange a crossing, replace a crossing like X with one like Y. In other words, an “under-crossing” becomes an “over-crossing”, and vice versa.

Starting with the original left-hand diagram, it is possible to obtain other links by interchanging crossings. There are many different ways of choosing which crossings to interchange.

Question 2
How many essentially different links can be obtained in this way?
You should explain what you take “essentially different” to mean.

Four loops

Question 3
Is there a link of four loops with the property that if any one of the loops is removed then the three remaining loops are unlinked?

Solutions and discussion

Question 1

Link diagram for left-hand picture Link diagram for right-hand picture

The most obvious difference between the two links is that the right-hand link falls apart — one of the loops (bottom left) can be separated from the other two — whereas in the Borromean rings the three loops are definitely all linked.

Almost as obvious a difference is that in the right-hand link two of the loops are linked whereas the Borromean rings have the property that removing any one loop always leaves two unlinked loops.

See the references below for more information about the Borromean rings and their history.

Question 2

Considering two links to be the same if one can be manipulated and deformed to look like the other (without breaking and joining the loops of string), we may obtain five essentially different links by interchanging crossings.

A full analysis of this problem may be found on Peter Cromwell's web page Borromean Rings - Theme and Variations.

Notice that there are many other different links with three loops, but none can be obtained by interchanging crossings in the Borromean rings diagram.

The Borromean rings

Question 3

There are many different ways of linking four loops so that if any one of the loops is removed then the three remaining loops are unlinked. One example is shown alongside; other examples can be found in the references below. Apart from the last loop, this link can be made in the form of a chain of rubber bands.

You should be able to see how to generalise the arrangement shown to give a link with the same property that has any number of loops.

Such links generalise the Borromean rings and are known as Brunnian links. See the references for more information.

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The United Kingdom Mathematics Trust organises the Maths Challenges and enrichment activities for schools and colleges.

United Kingdom Mathematics Trust
The Further Mathematics Network

The Network provides and coordinates enrichment for KS4 and sixth form students.

The Further Mathematics Network