7 Bridges

Jack Dikian

January 2011

While writing about Knot Theory last night the subject and associated problems in Topology bought back many memories that are reminiscent to a child experiencing the color and movement of a circus for the first time.

Topology and in particular Graph Theory is often not part of high school math curricula, and thus for many it sounds strange and intimidating. However, there are some readily graspable ideas at the base of topology that are interesting, fun, and highly applicable to all sorts of situations. One of these is the topology of networks, first developed by Euler in the early 1700’s inspired by the Seven Bridges of Konigsberg.

Konigsberg is a city which was the capital of East Prussia but now is known as Kaliningrad in Russia. The city is built around the River Pregel where it joins another river. An island named Kniephof is in the middle of where the two rivers join. There are seven bridges that join the different parts of the city on both sides of the rivers and the island.

The people wondered whether or not one could walk around the city in a way that would involve crossing each bridge exactly once. No one was able to find a way to do it, and this came to the attention of a mathematician Euler.

In 1735, Euler, before the Russian Academy presented the solution to the problem explaining why crossing all seven bridges without crossing a bridge twice was impossible.

Euler simplified the bridge problem by representing each land mass as a point and each bridge as a line. He reasoned that anyone standing on land would have to have a way to get on and off. Thus each land mass would need an even number of bridges. But in Konigsberg, each land mass had an odd number of bridges. This was why all seven bridges could not be crossed without crossing one more than twice.

Euler's recognition that the key information was the number of bridges and the list of their endpoints (rather than their exact positions) presaged the development of topology. The difference between the actual layout and the graph schematic is a good example of the idea that topology is not concerned with the rigid shape of objects.

Untangling Knots

Jack Dikian

January 2011

Recently, over the Christmas and New Year holidays, I found myself getting increasingly irritable with the number of times I found I needed to untangle my Ipod’s headphones. The angrier I become, the harder I tried to untangle the mess, the tighter the knots seemed to get. My young niece on the other hand approached the problem very differently. For her, it was a breeze - calm, and almost mechanical – little fingers might have been a benefit.

What did interest me however was the idea of whether a heuristic approach can be taken to untangle a knot. Two things just changed in my language. One that I used the word “knot” in the singular and secondly, implicitly, the ability to identify the knot-type and use a set of steps to unwind, untangle the knot.

We all know from experience how to create knots. We do this all the time, often, like me, unwittingly. In Pure Maths, we learned about those knots whose ends are glued together and their classification form the subject of a branch of Topology known as Knot Theory.

The thread goes back almost 150 years when in the nineteenth century physicists were speculating about the underlying principles of atoms. Lord Kelvin put forward a comprehensive theory of atoms, which, seemed to explain several of the essential qualities of the chemical elements - Kelvin's theory conjectured that atoms were knotted tubes of ether.

Kelvin's theory of atoms was taken seriously for many years. Maxwell for example thought that ``it satisfies more of the conditions than any atom hitherto considers''. The topological stability and the variety of knots were thought to mirror the stability of matter and the variety of chemical elements.

This theory inspired physicist Peter Tait to undertake an extensive study and tabulation of knots in an attempt to understand when two knots were ``different''. He’s intuitive understanding of ``different'' and ``same'' is a useful notion even today. Two knots are isotopic if one can be continuously manipulated in 3-space with no self-intersections allowed until it looks like the other.

Mathematicians, then and now, continue to pose the same question. That is, how do we know when two knots are isotopically the same. This failed atomic theory also left in its wake the riches of Tait's tabulation---163 knot projections---and a rudimentary understanding of isotopic sameness in terms of how one projection could be continuously manipulated to look like another. This understanding of projection manipulation was summarized in a set of conjectures for knot projections, the famous Tait Conjectures.