More Like this

1. Contact three-manifolds with exactly two simple Reeb orbits

   https://doi.org/10.2140/gt.2023.27.3801  

   Cristofaro-Gardiner, Daniel; Hryniewicz, Umberto; Hutchings, Michael; Liu, Hui (January 2023, Geometry & Topology)

   It is known that every contact form on a closed three-manifold has at least two simple Reeb orbits, and a generic contact form has infinitely many. We show that if there are exactly two simple Reeb orbits, then the contact form is nondegenerate. Combined with a previous result, this implies that the three-manifold is diffeomorphic to the three-sphere or a lens space, and the two simple Reeb orbits are the core circles of a genus-one Heegaard splitting. We also obtain further information about the Reeb dynamics and the contact structure. For example, the Reeb flow has a disk-like global surface of section and so its dynamics are described by a pseudorotation, the contact structure is universally tight, and in the case of the three-sphere the contact volume and the periods and rotation numbers of the simple Reeb orbits satisfy the same relations as for an irrational ellipsoid. 

   more » « less
2. On Newton polytopes of Lagrangian augmentations

   https://doi.org/10.1112/blms.12992  

   Capovilla‐Searle, Orsola; Casals, Roger (February 2024, Bulletin of the London Mathematical Society)

   Abstract This note explores the use of Newton polytopes in the study of Lagrangian fillings of Legendrian submanifolds. In particular, we show that Newton polytopes associated to augmented values of Reeb chords can distinguish infinitely many distinct Lagrangian fillings, both for Legendrian links and higher dimensional Legendrian spheres. The computations we perform work in finite characteristic, which significantly simplifies arguments and also allows us to show that there exist Legendrian links with infinitely many nonorientable exact Lagrangian fillings. 

   more » « less
3. Realizable Piecewise Linear Paths of Persistence Diagrams with Reeb Graphs

   https://doi.org/10.57717/CGT.V3I1.38  

   Alharbi, Rehab; Chambers, Erin; Munch, Elizabeth (July 2024, Computing in Geometry and Topology)

   Reeb graphs are widely used in a range of fields for the purposes of analyzing and comparing complex spaces via a simpler combinatorial object. Further, they are closely related to extended persistence diagrams, which largely but not completely encode the information of the Reeb graph. In this paper, we investigate the effect on the persistence diagram of a particular continuous operation on Reeb graphs; namely the (truncated) smoothing operation. This construction arises in the context of the Reeb graph interleaving distance, but separately from that viewpoint provides a simplification of the Reeb graph which continuously shrinks small loops. We then use this characterization to initiate the study of inverse problems for Reeb graphs using smoothing by showing which paths in persistence diagram space (commonly known as vineyards) can be realized by a path in the space of Reeb graphs via these simple operations. This allows us to solve the inverse problem on a certain family of piecewise linear vineyards when fixing an initial Reeb graph. While this particular application is limited in scope, it suggests future directions to more broadly study the inverse problem on Reeb graphs. 

   more » « less
4. Computing Reeb dynamics on four-dimensional convex polytopes

   https://doi.org/10.3934/jcd.2021016  

   Chaidez, Julian; Hutchings, Michael (January 2021, Journal of Computational Dynamics)

   null (Ed.)

   We study the combinatorial Reeb flow on the boundary of a four-dimensional convex polytope. We establish a correspondence between "combinatorial Reeb orbits" for a polytope, and ordinary Reeb orbits for a smoothing of the polytope, respecting action and Conley-Zehnder index. One can then use a computer to find all combinatorial Reeb orbits up to a given action and Conley-Zehnder index. We present some results of experiments testing Viterbo's conjecture and related conjectures. In particular, we have found some new examples of polytopes with systolic ratio \begin{document}$ 1 $$\end{document}$. 

   more » « less
5. An Efficient Data Retrieval Parallel Reeb Graph Algorithm

   https://doi.org/10.3390/a13100258  

   Hajij, Mustafa; Rosen, Paul (October 2020, Algorithms)

   null (Ed.)

   The Reeb graph of a scalar function that is defined on a domain gives a topologically meaningful summary of that domain. Reeb graphs have been shown in the past decade to be of great importance in geometric processing, image processing, computer graphics, and computational topology. The demand for analyzing large data sets has increased in the last decade. Hence, the parallelization of topological computations needs to be more fully considered. We propose a parallel augmented Reeb graph algorithm on triangulated meshes with and without a boundary. That is, in addition to our parallel algorithm for computing a Reeb graph, we describe a method for extracting the original manifold data from the Reeb graph structure. We demonstrate the running time of our algorithm on standard datasets. As an application, we show how our algorithm can be utilized in mesh segmentation algorithms. 

   more » « less