Low-dimensional Topology Research Articles

A space level light bulb theorem in all dimensions

Given a d -dimensional manifold M and a knotted sphere s\colon \mathbb{S}^{k-1}\hookrightarrow \partial M with 1\leq k\leq d , for which there exists a framed dual sphere G\colon \mathbb{S}^{d-k}\hookrightarrow \partial M , we show that the space of neat embeddings \mathbb{D}^{k} \hookrightarrow M with boundary s can be delooped by the space of neatly embedded (k-1) -disks, with a normal vector field, in the d -manifold obtained from M by attaching a handle to G . This increase in codimension significantly simplifies the homotopy type of such embedding spaces, and is of interest also in low-dimensional topology. In particular, we apply the work of Dax to describe the first interesting homotopy group of these embedding spaces, in degree d-2k . In a separate paper we use this to give a complete isotopy classification of 2-disks in a 4-manifold with such a boundary dual.

Chiral, Topological, and Knotted Colloids in Liquid Crystals

The geometric shape, symmetry, and topology of colloidal particles often allow for controlling colloidal phase behavior and physical properties of these soft matter systems. In liquid crystalline dispersions, colloidal particles with low symmetry and nontrivial topology of surface confinement are of particular interest, including surfaces shaped as handlebodies, spirals, knots, multi-component links, and so on. These types of colloidal surfaces induce topologically nontrivial three-dimensional director field configurations and topological defects. Director switching by electric fields, laser tweezing of defects, and local photo-thermal melting of the liquid crystal host medium promote transformations among many stable and metastable particle-induced director configurations that can be revealed by means of direct label-free three-dimensional nonlinear optical imaging. The interplay between topologies of colloidal surfaces, director fields, and defects is found to show a number of unexpected features, such as knotting and linking of line defects, often uniquely arising from the nonpolar nature of the nematic director field. This review article highlights fascinating examples of new physical behavior arising from the interplay of nematic molecular order and both chiral symmetry and topology of colloidal inclusions within the nematic host. Furthermore, the article concludes with a brief discussion of how these findings may lay the groundwork for new types of topology-dictated self-assembly in soft condensed matter leading to novel mesostructured composite materials, as well as for experimental insights into the pure-math aspects of low-dimensional topology.

Closed geodesics and Frøyshov invariants of hyperbolic three-manifolds

Frøyshov invariants are numerical invariants of rational homology three-spheres derived from gradings in monopole Floer homology. In the past few years, they have been employed to solve a wide range of problems in 3- and 4-dimensional topology. In this paper, we look at connections with hyperbolic geometry for the class of minimal L -spaces. In particular, we study relations between Frøyshov invariants and closed geodesics using ideas from analytic number theory. We discuss two main applications of our approach. First, we derive explicit upper bounds for the Frøyshov invariants of minimal hyperbolic L -spaces purely in terms of volume and injectivity radius. Second, we describe an algorithm to compute Frøyshov invariants of minimal L -spaces in terms of data arising from hyperbolic geometry. As a concrete example of our method, we compute the Frøyshov invariants for all spin ^{c} structures on the Seifert–Weber dodecahedral space. Along the way, we also prove several results about the eta invariants of the odd signature and Dirac operators on hyperbolic three-manifolds which might be of independent interest.

NGCN: Drug-target interaction prediction by integrating information and feature learning from heterogeneous network.

Drug-target interaction (DTI) prediction is essential for new drug design and development. Constructing heterogeneous network based on diverse information about drugs, proteins and diseases provides new opportunities for DTI prediction. However, the inherent complexity, high dimensionality and noise of such a network prevent us from taking full advantage of these network characteristics. This article proposes a novel method, NGCN, to predict drug-target interactions from an integrated heterogeneous network, from which to extract relevant biological properties and association information while maintaining the topology information. It focuses on learning the topology representation of drugs and targets to improve the performance of DTI prediction. Unlike traditional methods, it focuses on learning the low-dimensional topology representation of drugs and targets via graph-based convolutional neural network. NGCN achieves substantial performance improvements over other state-of-the-art methods, such as a nearly 1.0% increase in AUPR value. Moreover, we verify the robustness of NGCN through benchmark tests, and the experimental results demonstrate it is an extensible framework capable of combining heterogeneous information for DTI prediction.

On the Whitehead nightmare and some related topics

In this very fast survey I will start by reviewing my work on the QSF property in geometric group theory. While working on this topic, I met some infinitistic complications which I called the Whitehead nightmare. This has analogues in other areas of low-dimensional topology. Afterwards I will present some joint work in progress with Louis Funar and Daniele Otera, where we plan to show that all groups, and only finitely presented groups are concerned by this paper, can avoid the nightmare in question. And also that all groups have the GSC (geometric simple connectivity) property, much stronger than QSF. Both QSF and GSC are explained in the main text.

Morphisms in Low Dimensions

This workshop brought together experts on interrelated topics in low-dimensional topology, centred around the common theme of ‘morphisms’. Our goal was to improve community understanding of recent developments in the field and to promote new advances in the study of global properties of 4-manifolds.

Low-dimensional Topology

The workshop brought together experts from across all areas of low-dimensional topology, including knot theory, computational topology, three-manifolds and four-manifolds. In addition to the standard research talks we had two survey talks by Marc Lackenby and Joel Hass, leading to discussions of open problems. Furthermore we had three sessions of fiveminute talks by a total of roughly thirty participants.

A survey of the homology cobordism group

In this survey, we present the most recent highlights from the study of the homology cobordism group, with particular emphasis on its long-standing and rich history in the context of smooth manifolds. Further, we list various results on its algebraic structure and discuss its crucial role in the development of low-dimensional topology. Also, we share a series of open problems about the behavior of homology 3 3 -spheres and the structure of Θ Z 3 \Theta _{\mathbb {Z}}^3 . Finally, we briefly discuss the knot concordance group C \mathcal {C} and the rational homology cobordism group Θ Q 3 \Theta _{\mathbb {Q}}^3 , focusing on their algebraic structures, relating them to Θ Z 3 \Theta _{\mathbb {Z}}^3 , and highlighting several open problems. The appendix is a compilation of several constructions and presentations of homology 3 3 -spheres introduced by Brieskorn, Dehn, Gordon, Seifert, Siebenmann, and Waldhausen.

Graph Neural Networks and 3-dimensional topology

We test the efficiency of applying geometric deep learning to the problems in low-dimensional topology in a certain simple setting. Specifically, we consider the class of 3-manifolds described by plumbing graphs and use graph neural networks (GNN) for the problem of deciding whether a pair of graphs give homeomorphic 3-manifolds. We use supervised learning to train a GNN that provides the answer to such a question with high accuracy. Moreover, we consider reinforcement learning by a GNN to find a sequence of Neumann moves that relates the pair of graphs if the answer is positive. The setting can be understood as a toy model of the problem of deciding whether a pair of Kirby diagrams give diffeomorphic 3- or 4-manifolds.

The Jones polynomial, Knots, diagrams, and categories

This essay is a remembrance of Vaughan Jones and a diagrammatic exposition of the remarkable breakthroughs in knot theory and low-dimensional topology that were catalyzed by his work.

Topological Symmetry Groups of the Petersen Graphs

The topological symmetry group of an embedding Γ of an abstract graph γ in S3 is the group of automorphisms of γ that can be realized by homeomorphisms of the pair (S3,Γ). These groups are motivated by questions about the symmetries of molecules in space. The Petersen family of graphs is an important family of graphs for many problems in low-dimensional topology, so it is desirable to understand the possible groups of symmetries of their embeddings in space. In this paper, we find all the groups that can be realized as topological symmetry groups for each of the graphs in the Petersen family. Along the way, we also complete the classification of the realizable topological symmetry groups for K3,3.

Rigid and separable algebras in fusion 2-categories

Rigid monoidal 1-categories are ubiquitous throughout quantum algebra and low-dimensional topology. We study a generalization of this notion, namely rigid algebras in an arbitrary monoidal 2-category. Examples of rigid algebras include G-graded fusion 1-categories, and G-crossed fusion 1-categories. We explore the properties of the 2-categories of modules and of bimodules over a rigid algebra, by giving a criterion for the existence of right and left adjoints. Then, we consider separable algebras, which are particularly well-behaved rigid algebras. Specifically, given a fusion 2-category, we prove that the 2-categories of modules and of bimodules over a separable algebra are finite semisimple. Finally, we define the dimension of a connected rigid algebra in a fusion 2-category, and prove that such an algebra is separable if and only if its dimension is non-zero.

Group-valued invariant of knots in the full torus

Knot concordance plays a crucial role in the low-dimensional topology. We propose a very elementary techniques which allows one to construct a lot of sliceness obstructions for knots in the full torus. Our approach deals with group theoretical techniques; it is completely combinatorial, and the groups are very easy to deal with.

Renormalization in one-dimensional dynamics

The study of the dynamical and topological properties of interval exchange transformations and their natural generalizations is an important problem, which lies at the intersection of several branches of mathematics, including dynamical systems, low-dimensional topology, algebraic geometry, number theory, and geometric group theory. The purpose of the survey is to make a systematic presentation of the existing results on the ergodic and geometric characteristics of the one-dimensional maps under consideration, as well as on the measured foliations on surfaces and two-dimensional complexes that one can associate with these maps. These results are based on the research of the ergodic properties of the renormalization process - an algorithm that takes an original dynamical system and builds a sequence of equivalent dynamical systems with a smaller support set. For all dynamical systems considered in the paper these renormalization algorithms can be viewed as multidimensional fraction algorithms. Bibiliography: 74 titles.

Kindred diagrams

By a diagram we mean a topological space obtained by gluing to a standard circle a finite number of pairwise non-intersecting closed rectangles along their lateral sides, the glued rectangles are pairwise disjoint. Diagrams are not new objects; they were used in many areas of low-dimensional topology. Our main goal is to develop the theory of diagrams to a level sufficient for application in one more branch - in the theory of tangles. We provide the diagrams with simple additional structures - the smoothness of the circles and rectangles that are pairwise consistent with each other, the orientation of the circle, a point on the circle; we introduce new equivalence relation (that is as far as the author knows not previously encountered in the scientific literature) - kindred relation; we define a surjective mapping of the set of classes of kindred diagrams onto the set of classes of diffeomorphic smooth compact connected two-dimensional manifolds with boundary and note that in the simplest cases this surjection is also a bijection. The application of the constructed theory to the tangle theory requires additional preparation and therefore is not included in this article; the author intends to devote a separate publication to this application.

MATRIX-MFO Tandem Workshop: Invariants and Structures in Low-Dimensional Topology

The first ever MATRIX-MFO tandem workshop addressed several research questions in low-dimensional topology and related areas.

Categorical lifting of the Jones polynomial: a survey

This is a brief review of the categorification of the Jones polynomial and its significance and ramifications in geometry, algebra, and low-dimensional topology.

Symmetric Tangling of Honeycomb Networks

Symmetric, elegantly entangled structures are a curious mathematical construction that has found their way into the heart of the chemistry lab and the toolbox of constructive geometry. Of particular interest are those structures—knots, links and weavings—which are composed locally of simple twisted strands and are globally symmetric. This paper considers the symmetric tangling of multiple 2-periodic honeycomb networks. We do this using a constructive methodology borrowing elements of graph theory, low-dimensional topology and geometry. The result is a wide-ranging enumeration of symmetric tangled honeycomb networks, providing a foundation for their exploration in both the chemistry lab and the geometers toolbox.

The symplectic isotopy problem for rational cuspidal curves

We define a suitably tame class of singular symplectic curves in 4-manifolds, namely those whose singularities are modeled on complex curve singularities. We study the corresponding symplectic isotopy problem, with a focus on rational curves with irreducible singularities (rational cuspidal curves) in the complex projective plane. We prove that every such curve is isotopic to a complex curve in degrees up to five, and for curves with one singularity whose link is a torus knot. Classification results of symplectic isotopy classes rely on pseudo-holomorphic curves together with a symplectic version of birational geometry of log pairs and techniques from four-dimensional topology.

Sir Vaughan Jones. 31 December 1952—6 September 2020

Vaughan Jones made fundamental contributions to mathematics and mathematical physics, bringing together disparate areas of operator algebras, knots, links and low-dimensional topology in mathematics, and statistical mechanics, quantum field theory and quantum information in physics, while opening up the new field of quantum topology. The key that unlocked all this was his seminal work on subfactor theory in von Neumann algebras of operators, which led to a new invariant of links: the Jones polynomial. For this he was awarded the Fields Medal in 1990.