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1. Cluster Algebras and Its Applications

   https://doi.org/10.4171/OWR/2024/2  

   Baur, Karin; Marsh, Bethany; Schiffler, Ralf; Schroll, Sibylle (September 2024, Oberwolfach Reports)

   This workshop focused on recent developments in cluster algebras and their applications as well as interactions with other areas of mathematics. In addition to new advances in the theory of cluster algebras themselves, it included applications to knot theory and geometry as well as interactions with representation theory and categorification, Grassmannians, combinatorics, geometric surfaces models and Lie theory. 

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2. Topological approaches to knotted electric charge distributions

   https://doi.org/10.1007/s42985-023-00225-3  

   Lipton, Max (February 2023, Partial Differential Equations and Applications)

   Abstract Consider a knotKin$$S^3$$ $S^3$with uniformly distributed electric charge. Whilst solutions to the Laplace equation in terms of Dirichlet integrals are readily available, it is still of theoretical and physical interest to understand the qualitative behavior of the potential, particularly with respect to critical points and equipotential surfaces. In this paper, we demonstrate how techniques from geometric topology can yield novel insights from the perspective of electrostatics. Specifically, we show that when the knot is sufficiently close to a planar projection, we get a lower bound on the size of the critical set based on the projection’s crossings, improving a 2021 result of the author. We then classify the equipotential surfaces of a charged knot distribution by tracking how the topology of the knot complement restricts the Morse surgeries associated to the critical points of the potential. 

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3. Visualizing Mathematical Knot Equivalence

   https://doi.org/https://doi.org/10.2352/ISSN.2470-1173.2019.1.VDA-683  

   Juan Lin, Hui Zhang (January 2019, Electronic Imaging, Visualization and Data Analysis 2019)

   We present a computer interface to visualize and interact with mathematical knots, i.e., the embeddings of closed circles in 3-dimensional Euclidean space. Mathematical knots are slightly different than everyday knots in that they are infinitely stretchy and flexible when being deformed into their topological equivalence. In this work, we design a visualization interface to depict mathematical knots as closed node-link diagrams with energies charged at each node, so that highly-tangled knots can evolve by themselves from high-energy states to minimal (or lower) energy states. With a family of interactive methods and supplementary user interface elements, out tool allows one to sketch, edit, and experiment with mathematical knots, and observe their topological evolution towards optimal embeddings. In addition, out interface can extract from the entire knot evolution those key moments where successive terms in the sequence differ by critical change; this provides a clear and intuitive way to understand and trace mathematical evolution with a minimal number of visual frames. Finally out interface is adapted and extended to support the depiction of mathematical links and braids, whose mathematical concepts and interactions are just similar to our intuition about knots. All these combine to show a mathematically rich interface to help us explore and understand a family of fundamental geometric and topological problems. 

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4. Geometry Processing with Intrinsic Triangulations

   https://doi.org/10.1145/3450508.3464592  

   Sharp, Nicholas; Gillespie, Mark; Crane, Keenan (July 2021, SIGGRAPH '21: ACM SIGGRAPH 2021 Courses)

   This course provides a first introduction to intrinsic triangulations and their use in mesh processing algorithms. As geometric data becomes more ubiquitous, e.g., in applications such as augmented reality or machine learning, there is a pressing need to develop algorithms that work reliably on low-quality data. Intrinsic triangulations provide a powerful framework for these problems, by de-coupling the mesh used to encode geometry from the one used for computation. The basic shift in perspective is to encode the geometry of a mesh not in terms of ordinary vertex positions, but instead only in terms of edge lengths. Intrinsic triangulations have a long history in mathematics, but only in recent years have been applied to practical geometric computing. The course begins by giving motivation for intrinsic triangulations in terms of recent problems in computer graphics, followed by an interactive coding session where participants can make first contact with the idea of intrinsic meshes. We then give some mathematical background, and describe key data structures (overlay, signpost, normal coordinates). Using this machinery, we translate algorithms from computational geometry and scientific computing into cutting-edge algorithms for curved surfaces. For instance, we look at mesh parameterization, vector field processing, finding geodesics, solving partial differential equations (PDEs), and more. We also discuss processing of nonmanifold meshes and point clouds; participants can explore these algorithms via interactive demos. We conclude with a discussion of open questions and opportunities for future work. 

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5. Stabilization distance bounds from link Floer homology

   https://doi.org/10.1112/topo.12338  

   Juhász, András; Zemke, Ian (June 2024, Journal of Topology)

   Abstract We consider the set of connected surfaces in the 4‐ball with boundary a fixed knot in the 3‐sphere. We define the stabilization distance between two surfaces as the minimal such that we can get from one to the other using stabilizations and destabilizations through surfaces of genus at most . Similarly, we consider a double‐point distance between two surfaces of the same genus that is the minimum over all regular homotopies connecting the two surfaces of the maximal number of double points appearing in the homotopy. To many of the concordance invariants defined using Heegaard Floer homology, we construct an analogous invariant for a pair of surfaces. We show that these give lower bounds on the stabilization distance and the double‐point distance. We compute our invariants for some pairs of deform‐spun slice disks by proving a trace formula on the full infinity knot Floer complex, and by determining the action on knot Floer homology of an automorphism of the connected sum of a knot with itself that swaps the two summands. We use our invariants to find pairs of slice disks with arbitrarily large distance with respect to many of the metrics we consider in this paper. We also answer a slice‐disk analog of Problem 1.105 (B) from Kirby's problem list by showing the existence of non‐0‐cobordant slice disks. 

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