Links

Ben Laurie blathering


What are Ideal Knots and Links?

Earlier I wrote about an ideal link as if anyone had any clue what they were. So, here’s a brief explanation.

An ideal link is the form of the link that uses the least possible string. More formally, it is the shape that minimises L/r, where L is the total length of the centrelines and r is the radius of the thickest tube that can be put around that centreline without any overlapping (note that the same thickness of tube is used on all components of the link). Or, you can think of it as fixing the diameter of the tube and minimising L – intuitively, this is like tightening a piece of perfect string – which is why ideal links are also known as tight links.

So, why are they interesting? Well, apart from being very pretty, they predict the properties of knotted (or linked) DNA. The length of the ideal knot predicts very accurately the distance which the corresponding piece of knotted DNA travels when subjected to gel electrophoresis. They also predict the average writhe of a piece of knotted DNA that’s wriggling about in solution.

They also have some intrinsic properties that interest me. For example, they exhibit spontaneous symmetry breaking (the two components of that link are the same, topologically, but they end up different in the ideal link). Another is that, contrary to intuition, they don’t necessarily touch themselves everywhere – sometimes the shortest path is achieved by curving tighter than you can by wrapping around some other part of the knot, which means you have to break contact to do so.

Other people have other interests – for instance, my friend and collaborator John Maddocks is obsessed with the contact surface. Which, incidentally, I was recently given in 3D – aren’t 3D printers wonderful? I wish I had one.

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