Low dimensional topology

Low-dimensional topology is the study of 3- and 4-manifolds, and is closely linked with knot theory, where a knot is an embedding of the circle S^1 in S^3. Many invariants of knots have been defined over the past two centuries, and some of those are well-understood, but some are still mysterious. This project aims to get a better understanding of the latter, and to find new connections among these invariants.

The genus (or Seifert genus) g(K) of a knot K in S^3 is the least integer g such that K is the boundary of a connected, orientable 2-manifold M of genus g properly embedded in S^3.

The slice genus (or 4-ball genus) g_4(K) of K is the least integer g such that K is the boundary of a connected, orientable 2-manifold M of genus g properly embedded in the 4-ball B^4 bounded by S^3. Viewing this situation as M being in S^3x[0,1) with boundary equal to K in S^3x{0}, we can record a movie which shows the intersection of M and S^3x{t} at time t. This starts with K and deforms it, but at certain times we might also witness crossing changes, band surgeries, or new small circles appearing or disappearing.

We say that the knots K and K' are concordant if g_4(K#-K')=0. Concordance is an equivalence relation on the set of knots, and the equivalence classes form a group with inverse being the mirror of the knot. There is currently a lot of activity on trying to understand the structure of this group. For example, it is unknown if it contains torsion of order other than 2.

The unknotting number u(K) is the minimal number of crossing changes needed to transform K to the unknot. Currently, there is no algorithm for computing it. It is unknown whether the unknotting number is additive with respect to connected sums.

Currently, there is no algorithm for computing the slice genus or the unknotting number of a knot. It is well-known that g_4(K) is at most u(K), and obviously g_4(K) is at most g(K). My aim is to explore other relations between these invariants, find new bounds on the slice genus and the unknotting number, and better understand the structure of the concordance group.

Heegaard Floer homology is a package of invariants introduced by Ozsváth and Szabó, and Juhász defined cobordism maps induced on knot Floer homology. Twisted Alexander polynomials and various signature invariants have also been successfully used in the study of knot concordance. These powerful tools should allow me to make progress on the questions above.

An important application arises in biology: when a cell divides, the DNA is copied, and the copy wraps around the original many times. The role of enzymes such as topoisomerases, which act via a crossing change, and recombinases, acting via band surgery, is to untangle the two strands so that the copy can migrate into the daughter cell. Mathematically, one starts with a (2,2n) torus link or a satellite of it and ends up with a two-component unlink. Hence this is a direct analogue of watching the movie corresponding to the slice genus. I expect that a better understanding of the possible pathways from a two-component link to the unlink via crossing changes and band surgeries will shed light on the precise action of the enzymes.

This project falls within the EPSRC Geometry & Topology research area.

There are no companies or collaborators involved.