Periodic Reporting for period 4 - OXTOP (Low-dimensional topology in Oxford)
Reporting period: 2020-11-01 to 2021-10-31
Low-dimensional topology studies the shape of 3-dimensional and 4-dimensional spaces. Surprisingly, higher dimensional spaces are easier to understand as there is room to perform a certain topological trick. Hence dimensions 3 and 4 are the most difficult but also physically the most relevant.
Our Universe is 3-dimensional. We can only see a small segment of it limited by the range of our telescopes. This segment looks like 3-dimensional coordinate space, but globally it might have a more complicated shape. To better understand what we mean by that, imagine we are tiny short-sighted observers living in a 2-dimensional universe that is the surface of a doughnut. What we would see is indistinguishable from the 2-dimensional coordinate plane no matter where we stand, and we would find it difficult to decide whether our world is the plane, a sphere, or a doughnut. In case of the Earth, people observed its curvature to deduce it has the shape of a sphere.
If we also consider time, we obtain the concept of the 4-dimensional space-time, whose shape is a mystery. More generally, an n-manifold for any non-negative integer n is a space that locally looks like n-dimensional coordinate space. Topology is the study of the properties of such objects which are unchanged by continuous deformations, as if they were made out of rubber, but not in the local geometry such as its curvature. From the point of view of topology, the surface of the Earth and a sphere have the same shape: one can be continuously deformed into the other without tearing or puncturing it. To distinguish two spaces, topologists develop ingenious invariants that are unchanged by deformation. For example, the number of holes of a surface is a numerical invariant.
In dimension 3, the existence of knots becomes central due to the lack of room. There are many different ways one can tie a knot in a circular piece of rope. In fact, any 3- and 4-manifold can be described by a collection of such knots, each labeled by a number. 3-manifolds have been well understood using ideas from geometry, similarly to the way our predecessors realized the Earth is round by noticing its curvature.
However, the classification of 4-manifolds up to smooth deformations is still little understood. Here ideas from theoretical physics have been revolutionary. The most important open question is the smooth 4-dimensional Poincaré conjecture (SPC4): This states that if a 4-manifold can be continuously deformed into the standard 4-sphere, then this can also be done smoothly. Many experts believe this to be false, and to find a counterexample, it would suffice to show that a certain knot in 3-space is not the boundary of a disk in 4-space.
The first part of the project aims to develop tools sensitive enough to decide what kind of surfaces a given knot bounds in 4-dimensions, and use this to find a counterexample to SPC4. Surprisingly, this might also shed light on the action of certain enzymes called recombinases on the DNA which separates the copy of the DNA strand when a cell divides. In another direction, I am planning to use certain algebraic objects related to theoretical physics to define new invariants of 3- and 4-manifolds. Finally, in dimension 3, I will explore the relationship of a deep invariant of 3-manifolds and the geometries they possess.
Our Universe is 3-dimensional. We can only see a small segment of it limited by the range of our telescopes. This segment looks like 3-dimensional coordinate space, but globally it might have a more complicated shape. To better understand what we mean by that, imagine we are tiny short-sighted observers living in a 2-dimensional universe that is the surface of a doughnut. What we would see is indistinguishable from the 2-dimensional coordinate plane no matter where we stand, and we would find it difficult to decide whether our world is the plane, a sphere, or a doughnut. In case of the Earth, people observed its curvature to deduce it has the shape of a sphere.
If we also consider time, we obtain the concept of the 4-dimensional space-time, whose shape is a mystery. More generally, an n-manifold for any non-negative integer n is a space that locally looks like n-dimensional coordinate space. Topology is the study of the properties of such objects which are unchanged by continuous deformations, as if they were made out of rubber, but not in the local geometry such as its curvature. From the point of view of topology, the surface of the Earth and a sphere have the same shape: one can be continuously deformed into the other without tearing or puncturing it. To distinguish two spaces, topologists develop ingenious invariants that are unchanged by deformation. For example, the number of holes of a surface is a numerical invariant.
In dimension 3, the existence of knots becomes central due to the lack of room. There are many different ways one can tie a knot in a circular piece of rope. In fact, any 3- and 4-manifold can be described by a collection of such knots, each labeled by a number. 3-manifolds have been well understood using ideas from geometry, similarly to the way our predecessors realized the Earth is round by noticing its curvature.
However, the classification of 4-manifolds up to smooth deformations is still little understood. Here ideas from theoretical physics have been revolutionary. The most important open question is the smooth 4-dimensional Poincaré conjecture (SPC4): This states that if a 4-manifold can be continuously deformed into the standard 4-sphere, then this can also be done smoothly. Many experts believe this to be false, and to find a counterexample, it would suffice to show that a certain knot in 3-space is not the boundary of a disk in 4-space.
The first part of the project aims to develop tools sensitive enough to decide what kind of surfaces a given knot bounds in 4-dimensions, and use this to find a counterexample to SPC4. Surprisingly, this might also shed light on the action of certain enzymes called recombinases on the DNA which separates the copy of the DNA strand when a cell divides. In another direction, I am planning to use certain algebraic objects related to theoretical physics to define new invariants of 3- and 4-manifolds. Finally, in dimension 3, I will explore the relationship of a deep invariant of 3-manifolds and the geometries they possess.
There has been excellent progress on Branch 1. Project A is mostly complete. We have nearly finished subproject A1, and we are in the process of writing up the results. Zemke, who visited Oxford for two weeks on the grant, completed subproject A2 for link cobordisms. We showed his link cobordism maps for the hat version of HFL agree with my maps, and gave a handle theoretic description of the Honda-Kazez-Matic gluing maps. We have constructed exotic surfaces in the 4-ball, addressing Project A3.
In Project B (Finding a spectral sequence from Khovanov homology to knot Floer homology), the paper "Computing cobordism maps in link Floer homology and the reduced Khovanov TQFT" with Marengon provides a link between the Kohvanov TQFT and the link Floer TQFT. A spectral sequence has been constructed by Dowlin. We have made progress towards a simpler construction using the intersection model of the Jones and Alexander polynomials due to Palmer-Anghel.
Project C: Here, we have made outstanding progress. We have answered a question of Fintushel and Stern from the 90's on the effect of concordance surgery on the Heegaard Floer 4-manifold invariants, which is essentially the first original result on 4-manifolds obtained using Heegaard Floer theory. We have defined integer invariants of pairs of connected, properly embedded surfaces in the 4-ball bounding a knot in the 3-sphere that give lower bounds on the stabilization distance and the double point distance of the surfaces.
Celoria and Golla have also been working on Project C. In particular, they have made progress on subprojects C1 and C2.
The paper “Grid diagrams as tools to investigate knot spaces and topoisomerase-mediated simplification of DNA topology,” published in Science Advances, makes progress specifically in Project C4.
Together with Golla, we have shown that the EH class and the LOSS invariant of Legendrian knots in contact 3-manifolds are functorial under regular Lagrangian concordances in Weinstein cobordisms. This gives computable obstructions to the existence of regular Lagrangian concordances.
Branch 2 (TQFTs): There have been three PDRAs on the project working in the area: Andre Henriques, Bruce Bartlett, and Cristina Palmer-Anghel. Project E was studied by Banks. He constructed a (1+1)-TQFT whose state space agrees with that of the Rozansky-Witten TQFT. He also showed such a TQFT cannot be extended upwards to a (1+1+1)-TQFT. Hence, if the Rozanksy-Witten TQFT exists, it is not (1+1+1)-dimensional, answering Project E2. Banks also made progress on Project E1 in his PhD thesis.
Branch 4 (The Fox conjecture): Together with Peter Banks, we have started studying a graph-theoretic reformulation of the problem due to Kálmán. We have extended this from plane bipartite graphs to arbitrary directed graphs, and obtained computational evidence supporting this generalized conjecture. Our hope is to give an inductive proof of this more general conjecture, though the work is still in early stages.
The results have been disseminated in mathematical papers, all of which are also freely available on arxiv.org. We have organised a successful international workshop in Oxford that was attended by the leading experts in the field. The results were also communicated at international conferences and workshops, and in social media posts of the Mathematical Institute in Oxford.
In Project B (Finding a spectral sequence from Khovanov homology to knot Floer homology), the paper "Computing cobordism maps in link Floer homology and the reduced Khovanov TQFT" with Marengon provides a link between the Kohvanov TQFT and the link Floer TQFT. A spectral sequence has been constructed by Dowlin. We have made progress towards a simpler construction using the intersection model of the Jones and Alexander polynomials due to Palmer-Anghel.
Project C: Here, we have made outstanding progress. We have answered a question of Fintushel and Stern from the 90's on the effect of concordance surgery on the Heegaard Floer 4-manifold invariants, which is essentially the first original result on 4-manifolds obtained using Heegaard Floer theory. We have defined integer invariants of pairs of connected, properly embedded surfaces in the 4-ball bounding a knot in the 3-sphere that give lower bounds on the stabilization distance and the double point distance of the surfaces.
Celoria and Golla have also been working on Project C. In particular, they have made progress on subprojects C1 and C2.
The paper “Grid diagrams as tools to investigate knot spaces and topoisomerase-mediated simplification of DNA topology,” published in Science Advances, makes progress specifically in Project C4.
Together with Golla, we have shown that the EH class and the LOSS invariant of Legendrian knots in contact 3-manifolds are functorial under regular Lagrangian concordances in Weinstein cobordisms. This gives computable obstructions to the existence of regular Lagrangian concordances.
Branch 2 (TQFTs): There have been three PDRAs on the project working in the area: Andre Henriques, Bruce Bartlett, and Cristina Palmer-Anghel. Project E was studied by Banks. He constructed a (1+1)-TQFT whose state space agrees with that of the Rozansky-Witten TQFT. He also showed such a TQFT cannot be extended upwards to a (1+1+1)-TQFT. Hence, if the Rozanksy-Witten TQFT exists, it is not (1+1+1)-dimensional, answering Project E2. Banks also made progress on Project E1 in his PhD thesis.
Branch 4 (The Fox conjecture): Together with Peter Banks, we have started studying a graph-theoretic reformulation of the problem due to Kálmán. We have extended this from plane bipartite graphs to arbitrary directed graphs, and obtained computational evidence supporting this generalized conjecture. Our hope is to give an inductive proof of this more general conjecture, though the work is still in early stages.
The results have been disseminated in mathematical papers, all of which are also freely available on arxiv.org. We have organised a successful international workshop in Oxford that was attended by the leading experts in the field. The results were also communicated at international conferences and workshops, and in social media posts of the Mathematical Institute in Oxford.
As outlined above, we now have a very good understanding of the knot cobordism maps and how they can be used to detect exotic surfaces in 4-manifolds. We have understood the effect of concordance surgery, an operation on smooth 4-manifolds, on the Ozsvath-Szabo 4-manifold invariants. We have expanded our understanding of TQFTs and various quantum invariants, and the relationship between contact structures and Floer homology. As a novel tool, grid diagrams have been used for modelling the action of topoisomerases on DNA topology. Finally, we have answered a relative version of a problem on the Kirby problem list on 0-cobordisms.