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1. Parallelized Topological Relaxation Algorithm

   https://doi.org/10.1109/BigData47090.2019.9006309  

   Ruan, Guangchen; Zhang, Hui (December 2019, 2019 IEEE International Conference on Big Data (Big Data))

   Geometric problems of interest to mathematical visualization applications involve changing structures, such as the moves that transform one knot into an equivalent knot. In this paper, we describe mathematical entities (curves and surfaces) as link-node graphs, and make use of energy-driven relaxation algorithms to optimize their geometric shapes by moving knots and surfaces to their simplified equivalence. Furthermore, we design and conFigure parallel functional units in the relaxation algorithms to accelerate the computation these mathematical deformations require. Results show that we can achieve significant performance optimization via the proposed threading model and level of parallelization. 

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2. Topological Symmetry Groups of the Petersen Graphs

   https://doi.org/10.3390/sym15061267  

   Elzie, Deion; Fridhi, Samir; Mellor, Blake; Silva, Daniel; Wilson, Robin T. (June 2023, Symmetry)

   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. 

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3. Gompf’s Cork and Heegaard Floer Homology

   https://doi.org/10.1093/imrn/rnae180  

   Dai, Irving; Mallick, Abhishek; Zemke, Ian (August 2024, International Mathematics Research Notices)

   Abstract Gompf showed that for $$K$$ in a certain family of double-twist knots, the swallow-follow operation makes $1/n$-surgery on $$K \# -K$$ into a cork boundary. We derive a general Floer-theoretic condition on $$K$$ under which this is the case. Our formalism allows us to produce many further examples of corks, partially answering a question of Gompf. Unlike Gompf’s method, our proof does not rely on any closed 4-manifold invariants or effective embeddings, and also generalizes to other diffeomorphisms. 

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4. Toward Understanding and Evaluating Structural Node Embeddings

   https://doi.org/10.1145/3481639  

   Jin, Junchen; Heimann, Mark; Jin, Di; Koutra, Danai (June 2022, ACM Transactions on Knowledge Discovery from Data)

   While most network embedding techniques model the proximity between nodes in a network, recently there has been significant interest in structural embeddings that are based on node equivalences , a notion rooted in sociology: equivalences or positions are collections of nodes that have similar roles—i.e., similar functions, ties or interactions with nodes in other positions—irrespective of their distance or reachability in the network. Unlike the proximity-based methods that are rigorously evaluated in the literature, the evaluation of structural embeddings is less mature. It relies on small synthetic or real networks with labels that are not perfectly defined, and its connection to sociological equivalences has hitherto been vague and tenuous. With new node embedding methods being developed at a breakneck pace, proper evaluation, and systematic characterization of existing approaches will be essential to progress. To fill in this gap, we set out to understand what types of equivalences structural embeddings capture. We are the first to contribute rigorous intrinsic and extrinsic evaluation methodology for structural embeddings, along with carefully-designed, diverse datasets of varying sizes. We observe a number of different evaluation variables that can lead to different results (e.g., choice of similarity measure, classifier, and label definitions). We find that degree distributions within nodes’ local neighborhoods can lead to simple yet effective baselines in their own right and guide the future development of structural embedding. We hope that our findings can influence the design of further node embedding methods and also pave the way for more comprehensive and fair evaluation of structural embedding methods. 

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5. Visually Communicating Mathematical Knot Deformation

   https://doi.org/10.1145/3356422.3356438  

   Lin, Juan; Zhang, Hui (January 2019, VINCI'2019: Proceedings of the 12th International Symposium on Visual Information Communication and Interaction)

   Mathematical knots are different from everyday ropes in that they are infinitely stretchy and flexible when being deformed into their ambient isotopic. For this reason, a number of challenges arise when visualizing mathematical knot's static and changing structures during topological deformation. In this paper we focus on computational methods to visually communicate the mathematical knot's dynamics by computationally simulating the topological deformation and capturing the critical changes during the entire simulation. To further improve our visual experience, we propose a fast and adaptive method to extract key moments where only critical changes occur to represent and summarize the long deformation sequence. We conduct evaluation study to showcase the efficacy and efficiency of our methods. 

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