Low-dimensional Topology: Khovanov Homology and the Link with Heedaard-Floer Homology
Lead Research Organisation:
University of Glasgow
Department Name: School of Mathematics & Statistics
Abstract
Daniel Waite will use tools from Floer theory and 4-dimensional topology to attack various embedding and cobordism problems.
Our understanding of smooth 4-dimensional manifolds was revolutionised in the 1980s by Donaldson, who showed that solution spaces to differential equations coming from theoretical physics gave rise to new topological invariants. In the 1990s Seiberg and Witten gave a different set of equations, guided by dualities in physics, which were easier to work with and give more or less the same information as Donaldson invariants. In the 21st century these and related invariants have been applied to the study of 3-dimensional manifolds and also to knot theory -- the study of closed curves in 3-dimensional space. This has primarily been facilitated by the development of Heegaard Floer theory by Oszvath and Szabo; this gives a package of invariants which (by design) are a reformulation of Seiberg-Witten invariants, but which are more adapted to use in 3 dimensions and are more amenable to calculation.
Daniel will use various concordance and rational homology cobordism invariants, together with various 4-dimensional constructions, to obtain new results in two kinds of problem:
1) given a knot in the 3-sphere, what kind of embedded surfaces in the 4-dimensional ball can it bound?
2) given a 3-dimensional manifold, what kind of 4-dimensional manifolds can it bound, and what 4-manifolds can it be embedded into?
There are various relations between these problems, so progress in either one can lead to progress in the other.
This research will be of considerable interest to other mathematicians in the UK and worldwide working in gauge theory and low-dimensional topology. Problem 1) above is closely related to the study of Gordian distance between knots which is of interest to mathematical biologists studying knotted DNA molecules.
Our understanding of smooth 4-dimensional manifolds was revolutionised in the 1980s by Donaldson, who showed that solution spaces to differential equations coming from theoretical physics gave rise to new topological invariants. In the 1990s Seiberg and Witten gave a different set of equations, guided by dualities in physics, which were easier to work with and give more or less the same information as Donaldson invariants. In the 21st century these and related invariants have been applied to the study of 3-dimensional manifolds and also to knot theory -- the study of closed curves in 3-dimensional space. This has primarily been facilitated by the development of Heegaard Floer theory by Oszvath and Szabo; this gives a package of invariants which (by design) are a reformulation of Seiberg-Witten invariants, but which are more adapted to use in 3 dimensions and are more amenable to calculation.
Daniel will use various concordance and rational homology cobordism invariants, together with various 4-dimensional constructions, to obtain new results in two kinds of problem:
1) given a knot in the 3-sphere, what kind of embedded surfaces in the 4-dimensional ball can it bound?
2) given a 3-dimensional manifold, what kind of 4-dimensional manifolds can it bound, and what 4-manifolds can it be embedded into?
There are various relations between these problems, so progress in either one can lead to progress in the other.
This research will be of considerable interest to other mathematicians in the UK and worldwide working in gauge theory and low-dimensional topology. Problem 1) above is closely related to the study of Gordian distance between knots which is of interest to mathematical biologists studying knotted DNA molecules.
Organisations
People |
ORCID iD |
| Brendan Edward Owens (Primary Supervisor) | |
| Daniel Waite (Student) |
Studentship Projects
| Project Reference | Relationship | Related To | Start | End | Student Name |
|---|---|---|---|---|---|
| EP/N509668/1 | 01/10/2016 | 30/09/2021 | |||
| 1654027 | Studentship | EP/N509668/1 | 01/10/2015 | 03/09/2019 | Daniel Waite |
| Description | Concordance invariants of a family of pretzel knots have been determined through an algebraic construction recently proven to be equivalent to a family of knot invariants called Knot Floer Homology. These concordance invariants were previously unknown, and the author has also determined a related invariant for these pretzel knots, determinable for any member of the family. |
| Exploitation Route | This thesis highlights how one can extend a combinatorial construction with a computer implementation to determine invariants for a family of knots using inductive methods. |
| Sectors | Other |
| Description | Participant in 3MT (3 Minute Thesis Competition) |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | Local |
| Primary Audience | Postgraduate students |
| Results and Impact | The 3MT competition, run by the University of Glasgow, had the purposes of allowing researchers from across the University to concisely summarise their research for a general audience, with the aim of making research approachable and understandable within such a short time frame. The event was attended by approximately thirty people, and although this has not contributed to my research aims, it did stimulate discussion with other researchers, and videos of the talks were later made public online. |
| Year(s) Of Engagement Activity | 2017 |