Knot homology: theory and computation.

Knots to mathematicians are equivalent to what they are in real life: tangled up pieces of string. To a topologist it doesn't matter if you move the string around in space, so long as you keep the endpoints of the string fixed and you don't use your scissors! In order to distinguish between different knots, topologists compute quantities that don't change as you move the string around.

The research supported by this grant studies a particular class of these quantities (knot homologies) that have been discovered within about the last ten years. In doing this research we aim to prove theorems distinguishing between various knot homologies, and also carry out explicit computational work to understand better the structure of these theories. Interestingly, there are intimate connections of these theories with physics (gauge theory and string theory) so that we expect our research to have relevance even outside the realm of pure mathematics.

Impact Summary.

Most people are familiar with the double-helix structure of DNA. What many do not realize is that DNA does not just sit inside the nucleus of a cell as a long strand, but when it gets involved in cell processes it can get twisted up and knotted in the same way that a piece of string can be knotted.

A better analogy than a piece of string is the looped wire of a telephone cord, because this has a similar helical structure to DNA. In using a telephone, the handset can become rotated relative to the main body of the telephone, and this results in energy being stored in the cord. This energy naturally takes the form of what are known as supercoils or plectonemes. This is when two strands wrap around each other, giving rise to another "tertiary" helical structure. The same formation occurs in DNA: when taking part in cell processes, the ends of the DNA strand can become twisted relative to each other resulting in the presence of supercoils in the DNA. The presence of supercoiling is crucial to various biological processes.

In the field of topological biology, configurations of DNA are one of the fundamental objects of study. They are treated as knots and then studied using techniques of low-dimensional topology, recently including some arising from knot homologies (the topic of this grant). What makes this grant particularly interesting from the biological point of view is that the computational side of it concerns knots consisting of these supercoiled structures and so we expect to have particular relevance to topological studies of DNA.

In fact, supercoiling is the native state of DNA in every cell. So both in the cells and in the lab, DNA knots are almost always supercoiled. However, for the purposes of topological experimentation, one of the helices of the double helix is often chemically nicked, releasing the tension and thus removing the supercoiling. Consequently, a hope is that our methods could illuminate more realistic structures than those now detected.

Of course, the potential societal and economic impact of fundamental research into molecular biology is large. For example, understanding the topology of DNA interactions may lead to advances in understanding the genetic basis of predisposition to various medical disorders, for example cancer.