Taut foliations, representations, and the computational complexity of knot genus

This project involves the fields of topology (the mathematical study of shapes) and computational complexity (how to solve questions efficiently using a computer).

This project starts with the study of three-manifolds. A 'three-manifold' is a space that locally looks like the 3D space surrounding us. For example, imagine the complement of a closed, possibly tangled rope (i.e., a knot) inside 3D space. If we can continuously deform one knot into another one without tearing that first knot, we (topologists) consider the two knots to be the same knot. We can find a natural occurrence of such knots in our very own DNA structure, where the topological features of DNA (knots) reflect some inherited characteristics of the person they belong to.

Continuing from three-manifolds, we also look at 2D manifolds, known as 'surfaces', which locally look like geometric planes. Spheres and doughnuts (i.e., a torus, which has one handle) are the simplest examples of surfaces. If we were to remove a small disk from each of these surfaces, we'd get what we refer to as 'a surface with a boundary'. The 'genus' of this surface is the number of handles it has. Topologists have known that an important topological feature of a knot is indeed the 'knot genus'. We can define the genus of a knot (K) as the minimum genus between all, possibly tangled, orientable surfaces whose boundary coincides with K.

Determining the genus of a knot has been a very difficult question for quite some time. Agol, Hass and Thurston showed that if we allow both the knot and the ambient three-manifold to vary, then the question of `knot genus' is `NP-complete'.

The term 'NP-complete' deserves further explanation: There are many seemingly different questions across computational knowledge, from network theory to financial markets to internet security, all of which are actually equivalent from the pure mathematical angle. This means that if we have the solution to one of these questions, then we have the master key to unlock them all! Therefore, a solution to one NP-complete question is the gateway to a huge list of important answers across computational knowledge. This is one of the most fascinating beauties of mathematics: our work may unify these otherwise distant phenomena.

To this date, the only practical way of determining the knot genus involves what is known as the 'theory of foliations'. The theory's terminology is inspired by stratified rocks in geography, which gives a nice visual to the timeless nature and immensity of the kind of space we're talking about.

Moving forth, we understand a foliation of a three-manifold to be a partition of the three-manifold into surfaces (called 'leaves', the terminology being inspired by tree leaves), such that locally, the surfaces fit together no different than a stack of papers. The caveat here is that there can be infinite surfaces as well, something that we do not discuss here. A particularly important class of foliations are called `taut foliations'. Intuitively, a taut foliation has the property such that all its leaves minimize the area (like 'soap films', which are created when two soap bubbles merge and create a thin film between them).

The work of Agol, Hass and Thurston is also important for our understanding of the `P vs. NP question', a famous one that has puzzled computer scientists for decades. The P vs. NP is on the list of million-dollar Millennium Prizes by the Clay Institute, and it is the very basis of data encryption used by the public on a daily basis via the World Wide Web.

My proposed project aims to understand taut foliations and other related notions, and to continue the work of Agol, Hass and Thurston for furthering our understanding of the knot genus questions. This project will create new bridges between different areas of mathematics and computer science and can potentially have important applications to the study of DNA, and our understanding of the P vs NP question.

I propose to undertake fundamental research in mathematics that will interconnect several different areas of mathematics and computer science such as three-dimensional topology, contact topology, group theory and computational complexity. I will outline two different Pathways to Impact through which the proposed project can contribute to the scientific and technological advancements and the creation of wealth within the UK and more broadly in the world.

1) Technological advancements and wealth creation through data encryption and internet security: Cryptography and data protection are vital in the information age, as they are used on a daily basis in the financial markets and the World Wide Web. Therefore, it is of utmost importance to guarantee the safety of our encryption methods. The conjecture that the complexity classes P and NP are distinct is the basis of many cryptography methods. By the work of Agol-Hass-Thurston and Lackenby the unknot recognition now lies in both NP and co-NP but is not known to have a polynomial time algorithm. This was generalized by the work of Lackenby and I to `Determining knot genus in any fixed 3-manifold'. The current project builds on the work of Agol-Hass-Thurston to further our understanding of the unknot recognition and knot genus problems, which can shed new lights on the important P vs. NP question, or suggest new difficult questions as the basis of data encryption.


2) Scientific and technological advancement through applications of knot theory to the study of DNA: In the past decades, knot theory has been used for the study of the structure of DNA itself, and how DNA interacts with proteins (enzymes) in the cells. My project directly concerns with knots and in particular the unknot recognition problem, and can potentially lead to better algorithms for deciding whether a knot represents the unknot, and improved methods of computations for knot invariants such as knot genus. Historically taut foliations and contact structures have been successfully used to answer difficult questions about knots such as the Property R and Property P conjectures. Therefore, the systematic study of these structures in my project can help us in our understanding of knot theory which in turn can be used in the study of DNA as well as other tangled structures formed in the nature such as linked fluid vortices.