Computing the crosscap number of a knot using integer programming and normal surfaces

The crosscap number of a knot is an invariant describing the non-orientable surface of smallest genus that the knot bounds. Unlike knot genus (its orientable counterpart), crosscap numbers are difficult to compute and no general algorithm is known. We present three methods for computing crosscap number that offer varying trade-offs between precision and speed: (i) an algorithm based on Hilbert basis enumeration and (ii) an algorithm based on exact integer programming, both of which either compute the solution precisely or reduce it to two possible values, and (iii) a fast but limited precision integer programming algorithm that bounds the solution from above. The first two algorithms advance the theoretical state of the art, but remain intractable for practical use. The third algorithm is fast and effective, which we show in a practical setting by making significant improvements to the current knowledge of crosscap numbers in knot tables. Our integer programming framework is general, with the potential for further applications in computational geometry and topology.Comment: 19 pages, 7 figures, 1 table; v2: minor revisions; to appear in ACM Transactions on Mathematical Softwar

Enumerating fundamental normal surfaces: Algorithms, experiments and invariants

Computational knot theory and 3-manifold topology have seen significant breakthroughs in recent years, despite the fact that many key algorithms have complexity bounds that are exponential or greater. In this setting, experimentation is essential for understanding the limits of practicality, as well as for gauging the relative merits of competing algorithms. In this paper we focus on normal surface theory, a key tool that appears throughout low-dimensional topology. Stepping beyond the well-studied problem of computing vertex normal surfaces (essentially extreme rays of a polyhedral cone), we turn our attention to the more complex task of computing fundamental normal surfaces (essentially an integral basis for such a cone). We develop, implement and experimentally compare a primal and a dual algorithm, both of which combine domain-specific techniques with classical Hilbert basis algorithms. Our experiments indicate that we can solve extremely large problems that were once though intractable. As a practical application of our techniques, we fill gaps from the KnotInfo database by computing 398 previously-unknown crosscap numbers of knots.Comment: 17 pages, 5 figures; v2: Stronger experimental focus, restrict attention to primal & dual algorithms only, larger and more detailed experiments, more new crosscap number

Crosscap numbers and the Jones polynomial

We give sharp two-sided linear bounds of the crosscap number (non-orientable genus) of alternating links in terms of their Jones polynomial. Our estimates are often exact and we use them to calculate the crosscap numbers for several infinite families of alternating links and for several alternating knots with up to twelve crossings. We also discuss generalizations of our results for classes of non-alternating links.Comment: 27 pages. Minor corrections and modifications. To appear in Advances of Mathematic

Computational topology with Regina: Algorithms, heuristics and implementations

Regina is a software package for studying 3-manifold triangulations and normal surfaces. It includes a graphical user interface and Python bindings, and also supports angle structures, census enumeration, combinatorial recognition of triangulations, and high-level functions such as 3-sphere recognition, unknot recognition and connected sum decomposition. This paper brings 3-manifold topologists up-to-date with Regina as it appears today, and documents for the first time in the literature some of the key algorithms, heuristics and implementations that are central to Regina's performance. These include the all-important simplification heuristics, key choices of data structures and algorithms to alleviate bottlenecks in normal surface enumeration, modern implementations of 3-sphere recognition and connected sum decomposition, and more. We also give some historical background for the project, including the key role played by Rubinstein in its genesis 15 years ago, and discuss current directions for future development.Comment: 29 pages, 10 figures; v2: minor revisions. To appear in "Geometry & Topology Down Under", Contemporary Mathematics, AM

Computing Bounded Chains and Surfaces in a Simplicial Complex with Bounded-treewidth 1-skeleton

We consider three problems on simplicial complexes: the Optimal Bounded Chain Problem, the Optimal Homologous Chain Problem, and 2-Dim-Bounded-Surface. The Optimal Bounded Chain Problem asks to find the minimum weight d-chain in a simplicial complex K bounded by a given (d−1)-chain, if such a d-chain exists. The Optimal Homologous Chain problem asks to find the minimum weight (d−1)-chain in K homologous to a given (d−1)-chain. 2-Dim-Bounded-Surface asks whether or not there is a subcomplex of K homeomorphic to a given compact, connected surface bounded by a given subcomplex. All three problems are NP-hard, and the first two problems are hard to approximate within any constant factor assuming the Unique Games Conjecture. We prove that all three problems are fixed-parameter tractable with respect to the treewidth of the 1-skeleton of K

Interactive visualization tools for topological exploration

Thesis (Ph.D.) - Indiana University, Computer Science, 1992This thesis concerns using computer graphics methods to visualize mathematical objects. Abstract mathematical concepts are extremely difficult to visualize, particularly when higher dimensions are involved; I therefore concentrate on subject areas such as the topology and geometry of four dimensions which provide a very challenging domain for visualization techniques. In the first stage of this research, I applied existing three-dimensional computer graphics techniques to visualize projected four-dimensional mathematical objects in an interactive manner. I carried out experiments with direct object manipulation and constraint-based interaction and implemented tools for visualizing mathematical transformations. As an application, I applied these techniques to visualizing the conjecture known as Fermat's Last Theorem. Four-dimensional objects would best be perceived through four-dimensional eyes. Even though we do not have four-dimensional eyes, we can use computer graphics techniques to simulate the effect of a virtual four-dimensional camera viewing a scene where four-dimensional objects are being illuminated by four-dimensional light sources. I extended standard three-dimensional lighting and shading methods to work in the fourth dimension. This involved replacing the standard "z-buffer" algorithm by a "w-buffer" algorithm for handling occlusion, and replacing the standard "scan-line" conversion method by a new "scan-plane" conversion method. Furthermore, I implemented a new "thickening" technique that made it possible to illuminate surfaces correctly in four dimensions. Our new techniques generate smoothly shaded, highlighted view-volume images of mathematical objects as they would appear from a four-dimensional viewpoint. These images reveal fascinating structures of mathematical objects that could not be seen with standard 3D computer graphics techniques. As applications, we generated still images and animation sequences for mathematical objects such as the Steiner surface, the four-dimensional torus, and a knotted 2-sphere. The images of surfaces embedded in 4D that have been generated using our methods are unique in the history of mathematical visualization. Finally, I adapted these techniques to visualize volumetric data (3D scalar fields) generated by other scientific applications. Compared to other volume visualization techniques, this method provides a new approach that researchers can use to look at and manipulate certain classes of volume data

Generative Mesh Modeling

Generative Modeling is an alternative approach for the description of three-dimensional shape. The basic idea is to represent a model not as usual by an agglomeration of geometric primitives (triangles, point clouds, NURBS patches), but by functions. The paradigm change from objects to operations allows for a procedural representation of procedural shapes, such as most man-made objects. Instead of storing only the result of a 3D construction, the construction process itself is stored in a model file. The generative approach opens truly new perspectives in many ways, among others also for 3D knowledge management. It permits for instance to resort to a repository of already solved modeling problems, in order to re-use this knowledge also in different, slightly varied situations. The construction knowledge can be collected in digital libraries containing domain-specific parametric modeling tools. A concrete realization of this approach is a new general description language for 3D models, the "Generative Modeling Language" GML. As a Turing-complete "shape programming language" it is a basis of existing, primitv based 3D model formats. Together with its Runtime engine the GML permits - to store highly complex 3D models in a compact form, - to evaluate the description within fractions of a second, - to adaptively tesselate and to interactively display the model, - and even to change the models high-level parameters at runtime.Die generative Modellierung ist ein alternativer Ansatz zur Beschreibung von dreidimensionaler Form. Zugrunde liegt die Idee, ein Modell nicht wie üblich durch eine Ansammlung geometrischer Primitive (Dreiecke, Punkte, NURBS-Patches) zu beschreiben, sondern durch Funktionen. Der Paradigmenwechsel von Objekten zu Geometrie-erzeugenden Operationen ermöglicht es, prozedurale Modelle auch prozedural zu repräsentieren. Statt das Resultat eines 3D-Konstruktionsprozesses zu speichern, kann so der Konstruktionsprozess selber repräsentiert werden. Der generative Ansatz eröffnet unter anderem gänzlich neue Perspektiven für das Wissensmanagement im 3D-Bereich. Er ermöglicht etwa, auf einen Fundus bereits gelöster Konstruktions-Aufgaben zurückzugreifen, um sie in ähnlichen, aber leicht variierten Situationen wiederverwenden zu können. Das Konstruktions-Wissen kann dazu in Form von Bibliotheken parametrisierter, Domänen-spezifischer Modellier-Werkzeuge gesammelt werden. Konkret wird dazu eine neue allgemeine Modell-Beschreibungs-Sprache vorgeschlagen, die "Generative Modeling Language" GML. Als Turing-mächtige "Programmiersprache für Form" stellt sie eine echte Verallgemeinerung existierender Primitiv-basierter 3D-Modellformate dar. Zusammen mit ihrer Runtime-Engine erlaubt die GML, - hochkomplexe 3D-Objekte extrem kompakt zu beschreiben, - die Beschreibung innerhalb von Sekundenbruchteilen auszuwerten, - das Modell adaptiv darzustellen und interaktiv zu betrachten, - und die Modell-Parameter interaktiv zu verändern