Knots and networks
The mathematics of entanglement
Toen Castle believes we have much to learn (and gain) by subtlety modifying the connections between the nodes that make up a network, be that network a crystal lattice or a molecular material. His specific interest, which forms the basis of his PhD studies in the Department of Applied Maths (RSPSE), lies in understanding how networks of points can be entangled without changing the basic order of connection.
“This field of research started with people looking at networks that are embedded in space,” explains Mr Castle. “These approaches are relevant to a lot of physical systems such as crystals because crystal lattices can be thought of as networks of atoms and bonds in space.
“However, while quite a large amount of interest has been directed towards considering the connections between elements of these networks, not so much attention was given to the more subtle effects of the manner in which connections could be made.
“And when I looked at how connections might vary I began to concentrate on features that these bonds within the network can have. It’s possible to conceive of a tangling of complex structures, of things being connected in ways that are knotted or linked.
“A knot is a loop in space that you can’t straighten out into a normal circle in space without passing through itself. Knottedness is a fundamental property of a loop in space. Either it’s just a simple loop, like a circle, or else it’s got some form of structure in it that can never be removed without passing edges through each other.
“Links are related to knots. In the link there are more components and they’re joined together and unable to separate, which is a similar phenomenon, but involves two or more components instead of the one.”
Knots and links add a whole new dimension to the manner in which a network performs. You can have two networks with the exactly the same connectivity but which are linked or knotted in different ways making them behave in different ways.
A language of knots
“It’s very difficult to describe this area of tangled networks,” observes Mr Castle. “There’s no natural language to describe exactly what’s going on. You can wave your hands and say ‘look its tangled’, ‘look the layers are interlocked with each other’, but in terms of a quantitative science we’re really searching for a good description of what’s really going on. That doesn’t exist at the moment and that’s what I’m trying to do with my research.
“The aim is to come up with a framework or language by which this understanding of entanglement can be taken further and applied to different networks. And to be a good language, it must describe the fundamental tangling features that can be present in a network. So, by finding a conceptual language that can describe the tangling you can put together words from this language and generate novel structures.
“The language we’re generating is still rudimentary but we’ve made some real progress.”
To build a language of entanglement, a language of network knots, you ideally begin with a basic structure and explore what’s possible in terms of entangling it without changing its connectivity. And for Toen Castle, that structure is a cube.
“Most people conceive of a cube as a solid square block with six sides,” says Mr Castle. “For our purposes, ignore its volume and surface and think just of the edges as a wire frame with eight corners. The cube is a very interesting structure because it’s so common and hence we have a good intuitive link with it. However, it’s also complicated enough to start to display some interesting properties in terms of the potential ways to tangle it.
“It’s also a common building block in crystal lattices. In complex structures, modern crystallographers may find certain repeating cubic and tetrahedral units inside the crystal structure and then represent the structure in terms of those cubes and tetrahedra or prisms just to simplify their work.”
To understand how you might tangle a cube it’s helpful to consider the impossible cube. It has the same connectivity as your normal cube and yet it’s completely different. It’s tangled.
“Many people will have seen the impossible cube in books of optical illusions,” says Mr Castle. “It’s an optical enigma in which the lines and perspectives seem wrong. It seems impossible because there’s no way you could embed this version of the cube onto a sphere. By that I mean that there is no way to deform the wire-frame cube onto the surface of a sphere. The cube embedded on a sphere - a normal cube - can’t have any knots or links.
“If you consider every way of starting at a certain vertex of the cube, travelling along edges and coming back to your original vertex without doubling up, there’s no way that you can make a cycle that will be anything other than a simple loop; there’s no way it can have a knot in it, being embedded on the sphere just prohibits it. In terms of tangling, the sphere isn’t an exciting place to live. Similarly, there’s no way to find two distinct cycles that actually link together, they’re always just on opposite sides of the sphere.
“However, it’s quite easy to embed an impossible cube onto the surface of a doughnut. Another way of saying that is that there’s only one way you can embed a cube into a blob but there are many interesting ways you can embed that cube into a blob with a hole in it (that is, embed it into a doughnut or torus). And this is the beginning of how you build your language of entanglement.”
Knots beyond the doughnut
Having worked out a method to tangle basic structures by embedding them in doughnuts, Mr Castle is now looking to explore more complex entanglements.
“The next step in this process is to step up from the simple donut, a torus with one hole, to more complicated shapes like a donut with many holes,” says Mr Castle. “This is a very big challenge because the mathematics of multiple-holed donuts is quite different to single holed donuts.
“You get a vastly more structural complexity when you use multiple-holed donuts. You can chop them up and peel them open into repeating units in any number of ways giving you very interesting twisting and tangling patterns.”
While conceiving a language of entanglement might sound a little abstract, Mr Castle is a firm believer in the real world application of this work.
“While this work engages with some very sophisticated concepts, there’s real potential for applying this work,” he says. “Possible applications of this work include the creation of new crystal structures and new materials.”
“Hydrogen storage is another big application of this work because hydrogen, being a gas, takes a lot room to store but hydrogen has an affinity to stay at the surface of some materials. So, some mineral structures like zeolites, have lots of big rings in their structure, and are excellent for hydrogen storage. Materials engineers are trying to improve this storage structure and scale it up, however when they do this, the rings are prone to collapse around each other – and entangle. So, researchers are now looking to build similar structures that have the same features, but which are more stable and avoid tangling.
“And there are other ideas as well for how we might employ tangled networks and structures. Though I suspect that the best ideas on how to use this knowledge haven’t even occurred to us yet.”