  # Low-dimensional topology in Oxford

## Periodic Reporting for period 3 - OXTOP (Low-dimensional topology in Oxford)

Reporting period: 2019-05-01 to 2020-10-31

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.

In more technical terms, this project aims to build a group that brings together experts in gauge-theoretic, geometric, and group-theoretic techniques. It consists of 4 branches:

1. Cobordism maps in knot Floer homology (HFK). Defined by the PI, these should yield invariants of surfaces in 4-manifolds. Hence, they could be used to bound the 4-ball genus and the unknotting number, providing a tool for finding a counterexample to the smooth 4-dimensional Poincaré conjecture, and to decide whether a given slice knot bounds a ribbon surface. The cobordism maps seem to yield a spectral sequence from Khovanov homology to HFK. An important biological application is an obstruction for two links to be related by a band surgery.

2. Topological Quantum Field Theories (TQFTs). We use our classification of (2+1)-dimensional TQFTs in terms of J-algebras to find new examples of such TQFTs. First, we simplify the algebraic structure, then determine when a GNF*-algebra corresponds to a (1+1+1)-dimensional TQFT. This would allow us to find a (2+1)-dimensional TQFT that is not (1+1+1)-dimensional.

3. Heegaard Floer (HF) homology and geometrization. There are currently few links known between Floer-theoretic invariants of 3-manifolds and the geometric structures they admit. We propose to study the Floer homology of arithmetic 3-manifolds. These are often L-spaces; the question is when this happens, and whether the HF correction terms contain any number-theoretic information. The next step is studying the relationship between HF and the Thurston geometries, and then gluing along tori via bordered Floer homology. An important step is to understand the behaviour of HF under covering maps.

4. The Fox conjecture. This states that the absolute values of the coefficients of the Alexander polynomial of an alternating knot form a unimodal sequence. We propose a strategy for attacking this conjecture via the graph-theoretic description of the Alexander polynomial due to Kálmán, and the test of log-concavity of Huh.
"There has been excellent progress on Branch 1 (Cobordism maps in knot Floer homology). Project A (Strengthening the cobordism maps) is mostly complete. With Ghiggini and Zemke, we have nearly completed subproject A1 (Defining canonical orientation systems in sutured Floer homology), and we are in the process of writing up the results. Zemke, who visited Oxford for two weeks on the grant, completed subproject A2 (Cobordism maps on the plus and minus versions of sutured Floer homology) for link cobordisms. In ""Contact handles, duality, and sutured Floer homology"", 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. This allowed us to show that the triangle maps in SFH agree with the cobordism map induced by the sutured cobordism coming from the triple diagram. Together with my work ""Computing cobordism maps in link Floer homology and the reduced Khovanov TQFT"" with Marengon, this also gives a method of computing the link cobordism maps for elementary cobordisms. We have attempted to carry out subproject A3 (constructing mixed invariants for closed surfaces), but came to the conclusion this might be technically not feasible.

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. The difficulty is that each component of a knot in the link Floer TQFT needs to have at least two basepoints, hence the rank is too big to obtain a spectral sequence from knot Floer homology to Khovanov homology.

Project C (Applications of knot cobordism maps on surfaces in 4-manifolds): Here we have made outstanding progress. The above trace formula with Zemke has allowed us to answer 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 also managed to compute the invariant of deform-spun slice disks in knot Floer homology, and used this to distinguish infinitely many slice disks with diffeomorphic complements. Also with Zemke, 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. The stabilization distance of S and S' is the minimal g such that there is a sequence of surfaces connecting S and S' of genus at most g such that consecutive terms are related by a generalized stabilization or destabilization operation. The double point distance of S and S' of the same genus is defined as the minimum of the maximal number of double points appearing in regular homotopies from S to S'.

Celoria and Golla have also been working on Project C. In particular, they have made progress on subprojects C1 and C2. They have obtained results on (stable) concordance of knots and links, building on ideas of Hedden and Kuzbary. They applied techniques coming from Heegaard Floer homology (namely, correction terms with twisted coefficients) as well as more classical tools (e.g. triple cup products) to give obstructions to sliceness and concordance. They have also been studying concordance of knots in S^1 x S^2, and how it is related to the minimal geometric winding number. Their work is available in the preprint ""Heegaard Floer homology and concordance bounds on the Thurston norm"". Celoria, together with Alfieri and Stipsicz, has extended the upsilon concordance invariant to null-homologous knots in rational homology 3-spheres. By considering m-fold cyclic branched covers with m a prime power, this provides new knot concordance invariants; see their preprint ""Upsilon invariants from cyclic branched covers"".

In ""A note on cobordisms of algebraic knots"", Celoria, Bodnar, and Golla studied smooth cobordisms of algebraic knots and complex deformations of cusp singularities of curves using the invariant nu^+. They understood its behaviour with respect to connected sums, providing an explicit formula in the case of L-space knots and proving subadditivity in general.

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.

In Branch 2 (TQFTs), the foundational paper, Defining and classifying TQFTs via surgery, has appeared in Quantum Topology. There have been three PDRAs on the project working in the area: Andre Henriques, Bruce Bartlett, and Cristina Palmer-Anghel (who has just joined the group). Relating to subproject E2, my PhD student Peter Banks is trying to use my machinery to give a mathematically rigorous definition of the Rozansky-Witten TQFTs, which would provide examples of non-extendable (2+1)-dimensional TQFTs.

Henriques proved that Z(Rep(based loop group)) = Rep(free loop group). This is a major achievement, and the paper announcing this result has been published in PNAS. His result is relevant for operator algebras, for representation theory, and for mathematical physics.

From the point of view of operator algebras, Rep(based loop group) is a higher categorical analogue of a von Neumann algebra. The phenomenon of finding higher categorical analogues of existing mathematical notions is by now well-known, and has always lead to exciting new mathematics. From the point of view of representation theory, it is important to note that the representation theory of the based loop groups had never been considered before. The fact that the fusion product makes sense for these representations is remarkable. Finally, and most importantly, from the point of view of mathematical physics, this answers a question that had been around for a long time: “What does Chern-Simons theory assign to a point?” The answer is: Rep(based loop group).

My PhD student Kang has made progress on subproject G1 (behaviour of Heegaard Floer homology under finite covers) of Project 3. He proved that Z_2-equivariant HF is natural and functorial under based cobordisms in the case of knots. He has also constructed an invariant of transverse links in the standard contact 3-sphere, as a well-defined element of the equivariant Heegaard Floer cohomology of the branched double cover, which is functorial under constructible symplectic link cobordisms, and whose behaviour is similar to that of the LOSS invariant in HFK theory. In ""Z_2-equivariant Heegaard Floer cohomology of knots in S^3 as a strong Heegaard invariant"", he has shown that the Z_2-equivariant HF of a knot in the 3-sphere, constructed by Hendricks, Lipshitz, and Sarkar, is natural. Using this, he defined a transverse knot invariant that is a common refiniement of the LOSS invariant and the Z_2-equivariant contact class of transverse knots.

With Kang, we have also extended the definition of algebraic torsion, due to Kutluhan et al., to contact manifolds with convex boundary. The definition is in terms of a partial open book decomposition of the underlying sutured manifold supporting the contact structure, and an arc basis of the page. As a result, we have proved that the algebraic torsion of a codimension zero contact submanifold gives an upper bound on the algebraic torsion of the ambient manifold. This result can be used to show that if a contact manifold with convex boundary is overtwisted, then it has algebraic torsion zero, and if the given manifold has Giroux 2-torsion, then it has algebraic torsion at most two.

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."
As outlined above, we now have a very good understanding of the knot cobordism maps. We expect these to give further insight into surfaces in 4-manifolds, better understand the types of surfaces a knot can bound in the 4-ball (sliceness, 4-ball genus), and could potentially be used to detect exotic smooth 4-manifolds. We have started to study concordance of knots in 3-manifolds other than the 3-sphere.

The advances in our understanding of (2+1)-dimensional TQFTs might make it possible to give mathematically rigorous constructions of physical theories, such as Rozansky-Witten theory. The work of Henriques has important implications to operator algebras, representation theory, and mathematical physics.

The results of Kang provide potentially more sensitive invariants of transverse knots in equivariant Floer homology that are functorial under constructible symplectic cobordisms. This could have applications to contact and symplectic topology.

In the future, we shall focus also on branches 3 and 4 of the project to find further links between geometric 3-manifold topology and Heegaard Floer homology, and to attack the Fox conjecture on alternating knots.