Marie Trin: Train-tracks on surfaces

Train-tracks are some specific types of graphs on surfaces. Their study has been pioneered by William Thurston with the idea to make the study of measured laminations combinatorial. The goal of this introduction is to understand what train-tracks are and how useful they can be to study geometrical objects on surfaces: curves and measured laminations. We will start by focusing on the case of the torus and see how studying train-tracks can help to compute continued fraction expansions of real numbers. This will be our stepping stone towards measured laminations on surfaces and their encoding through train tracks.

Ian Moffat: From knots to ribbon graphs to delta-matroids

This is a gentle introduction to the topic of ribbon graphs in topological graph theory and their connections with knot theory. The route we take will follow a natural development of ideas. We'll take as our starting point the classic connection between the Jones polynomial of a knot and the Tutte polynomial of a graph. From this we'll see how removing restrictions from this connection takes us to the world of ribbon graphs. We'll then use the knot theory to as an anchor as a develop ribbon graph theory, covering topics such as graph polynomials, duals, minors, delta-matroids and multimatroids. The aim is that the audience will come away with enough background to easily digest the literature and pursue new ideas.

Erica Flapan: Spatial graph theory

We present the history of spatial graph theory from its roots in the study of chirality of flexible molecules to its ties to knot theory.  We explain intrinsic properties of spatial graphs including intrinsic knotting and linking and intrinsic chirality.  We then discuss symmetry and asymmetry of spatial graphs, including nonrealizable graph automorphisms and topological symmetry groups.  We explain the Simon invariant and its generalizations and applications.  Finally, we present recent results on projections of spatial graphs.

Federica Gavazzi: A Deligne Complex for Virtual Artin Groups

Artin groups can be viewed as generalizations of braid groups. Similarly, virtual Artin groups, introduced by Bellingeri, Paris, and Thiel, can be regarded as generalizations of virtual braid groups. Many natural questions about Artin groups remain open and are often studied using geometric methods, in particular through actions on non-positively curved spaces. In this talk, we will consider an analogue of the Deligne complex for virtual Artin groups and show that it is CAT(0) for a certain subclass of defining graphs. As a consequence of this action, we classify the finite subgroups of locally reducible virtual Artin groups, showing that, up to conjugacy, they are contained in the corresponding copy of the Coxeter group.

Nico Van  Cleemput: Combinatorial tiling theory

In the 1980s, Andreas Dress introduced what are now known as Delaney-Dress symbols, building on inspiration from work of M. S. Delaney. This work gave rise to combinatorial tiling theory: a framework for describing periodic tilings not through coordinates and geometry, but through a finite, purely combinatorial object -- a small colored graph that captures how vertices, edges, and tiles fit together, and how a tiling's symmetries act on them. Two tilings turn out to be equivariantly equivalent exactly when their Delaney-Dress symbols are isomorphic, turning a hard geometric classification problem into a tractable combinatorial one.

This talk gives a practical introduction to the theory. We build up the Delaney-Dress symbol from a tiling step by step, try to convey the intuition behind chambers, involutions, and branching numbers, and show how the symbol can be read back to recover a tiling's structure. Along the way we'll look at examples and tools used in practice, aiming to leave the audience equipped to work with Delaney-Dress symbols directly, rather than just recognizing the name.

Jo-Ellis Monaghan: When Is a Graph O.K.? Origami Knotting in Spatial Graphs

Authors:  Jo Ellis-Monaghan*,  Wout Moltmaker, Ada Morse, Greta Pangborn

Emergent technologies in DNA self-assembly present challenging design problems that have given rise to a new branch of mathematics, DNA mathematics.  Often, the self-assembled objects, e.g. lattices, polyhedral skeletons, or wireframe constructs, may be modeled as spatial graphs.   One design process, DNA origami, requires a route through the spatial graph for a scaffolding strand of DNA.  This route must trace an Eulerian circuit with restricted turnings at the vertices, which, when viewed as a space curve, is ideally unknotted.  Prior work addressed this problem via A-trails in graphs on surfaces embedded in R^3.  Here, we introduce new theory for arbitrary spatial graphs and origami trails (O-trails) as analogs of A-trails, and analyze their knotting behavior.  We show that origami knotting in Euler circuits is fundamentally different from previously studied intrinsic knotting of (Hamiltonian) cycles in graphs, particularly in that flatness does not suffice to assure unknottedness.   We show however that for 4-regular graphs, a strengthening of flatness does suffice to assure unknotted O-trails.  We also leverage prior work of  S. Barthel and F. Buccoliero on leveled surfaces to compare alternative approaches to the scaffolding route problem.  We close with a rich collection of open problems.

Sofia Lambropoulou: Algebraic and combinatorial methods for certain spatial structures 

We shall discuss various forms of spatial structures, such as bonded knots/knotoids, singular knots/knotoids, embedded graphs, and their topological equivalences. We will then sample some algebraic and combinatorial methods for measuring their topological entanglement. These include their connection to related braid-like structures, such as bonded braids/braidoids, singular braids/braidoids, as well as combinatorial invariants. We shall also brief on the various forms in the context of doubly periodic entanglements and equivalences.

Finn Thompson: The graph of grid moves as a knot invariant

Motivated by the monotonic simplification of grid diagrams of unknots due to Dynnikov, we explore the sequences of moves between two grid diagrams of the same knot. We describe the graph associated to grid diagrams of each knot type, where vertices represent grid diagrams, and are connected by an edge if they are related by a grid move. We determine a colouring of the edges that allows us to recognise the moves performed along edges. By analysing particular cycles and walks in a finite section of the graph, we aim to reconstruct a minimal grid diagram of the knot.

Mathijs Wintraecken: The Singular Source of Vineyard Monodromy

Vineyards, or time-varying families of persistence diagrams, are widely used in topological data analysis (TDA) pipelines to track how topological features change and evolve as a parameter varies.  When the parameter traces a closed loop, a vineyard can exhibit monodromy: diagram points permute over the course of a full traversal, which obstructs feature tracking and can complicate downstream analysis of such data.

Chambers et al. considered the periodic vineyards that arise from the radial persistence transform, which maps the manifold to a family of persistence diagrams, where each diagram fixes a base point and considers the filtration that is based on Euclidean distance to that point, and showed that monodromy and knotting can occur. Other recent work by Arya et al. considers geometric conditions that exclude monodromy in two dimensions, in an effort to better understand when this effect happens.

That said, understanding when and why monodromy occurs is a fundamental open problem with direct practical consequences for many data analysis pipelines.  

In this work, we study this question for 1-manifolds in R2, using a surprising connection with tools from singularity theory, and provide a classification for the causes of monodromy in vineyards.  More precisely, we prove that the vineyard of a sufficiently small loop \gamma cannot exhibit monodromy unless it contains a specific singularity of the distance function.  The central geometric object in our analysis is the symmetry set, which is the locus of centers of spheres tangent in more than one point to the manifold; this object classifies singularities of the distance function, and in our setting, dictates precisely when monodromy occurs.  This characterization opens the door to the development of algorithmic criteria for detecting and utilizing (or avoiding) monodromy in TDA pipelines.

Joint work with: Erin Chambers, Christopher Fillmore, Shankha Shubhra Mukherjee, Rohit Roy, and Elizabeth Stephenson.

Wout Moltmaker: Knotoids, Linkoids, and Periodic Tangles

A periodic tangle is an infinite system of linked curves with a translational symmetry; think of a chain link fence for example. More interestingly, a large class of three-dimensional periodic tangles is found in DNA tensegrity crystals. In this talk, we will consider a toy model consisting of two-dimensional periodic tangles, modelling things like fabrics or weaves. For these, there is a known relation to virtual links: modding out by the translational symmetry of a 2-periodic tangle one obtains a link on the torus, which can be seen as a virtual link. In this talk we go one step further and also consider the other wallpaper groups. As we shall see, in the identification space this brings us to several fun generalizations of knots including knotoids, linkoids, and twisted links on punctured surfaces, as well as novel equivalence relations on them.

Boudewijn Bosch: Perturbed Alexander Invariants for Linkoids

In recent work, Dror Bar-Natan and Roland van der Veen introduced fast and strong knot invariants that can be interpreted as perturbations of the Alexander invariant. In ongoing research, in collaboration with Wout Moltmaker and Kasturi Barkataki, we propose an extension of these invariants to linkoids, which can be thought of as link diagrams that admit open-ended components whose endpoints may lie in any region of the diagram. In line with work of Eleni Panagiotou and Kasturi Barkataki, this enables us to define a perturbed Alexander function for a collections of open and closed curves in 3-space. For collections of open curves, this function is a polynomial with real coefficients, depending continuously on the curves' coordinates. We hope this work could contribute to the study of entanglement in protein and polymer structures.

Davide Proserpio: Periodic Structures and Crystal Chemistry, A history of the topological approach to crystal chemistry by means of the cds net. 

In the rationalization process of complex structures it is very helpful the “topological approach” that consists in the simplification by rational principles of the complex structures to schematized reference nets.  A higher level of complexity comes from the entanglement of different periodic motifs whose rationalization and classification we have illustrated in the last years. The different branches of mathematics involved show the interdisciplinary approach of the Topological Crystal Chemistry. Examples will be illustrated with the free software ToposPro (topospro.com)