A Linear Time Algorithm for Visualizing Knotted Structures in 3 Pages

We introduce simple codes and fast visualization tools for knotted structures in molecules and neural networks. Knots, links and more general knotted graphs are studied up to an ambient isotopy in Euclidean 3-space. A knotted graph can be represented by a plane diagram or by an abstract Gauss code. First we recognize in linear time if an abstract Gauss code represents an actual graph embedded in 3-space. Second we design a fast algorithm for drawing any knotted graph in the 3-page book, which is a union of 3 half-planes along their common boundary line. The running time of our drawing algorithm is linear in the length of a Gauss code of a given graph. Three-page embeddings provide simple linear codes of knotted graphs so that the isotopy problem for all graphs in 3-space completely reduces to a word problem in finitely presented semigroups

Three-page encoding and complexity theory for spatial graphs

We construct a series of finitely presented semigroups. The centers of these semigroups encode uniquely up to rigid ambient isotopy in 3-space all non-oriented spatial graphs. This encoding is obtained by using three-page embeddings of graphs into the product of the line with the cone on three points. By exploiting three-page embeddings we introduce the notion of the three-page complexity for spatial graphs. This complexity satisfies the properties of finiteness and additivity under natural operations.Comment: 32 pages with 9 figures, submitted to J.Knot Theory and Ra

A Linear Time Algorithm for Drawing a Graph in 3 Pages within its Isotopy Class in 3-Space

We consider undirected graphs up to an ambient isotopy in 3-space. Such a graph can be represented by a plane diagram or a Gauss code. We recognize in linear time if a Gauss code represents an actual graph in 3-space. We also design a linear time algorithm for drawing a topological 3-page embedding of a graph isotopic to a given graph

Musubime to karamime no gurafu hyogen

制度:新 ; 報告番号:乙2277号 ; 学位の種類:博士(理学) ; 授与年月日:2010/6/22 ; 早大学位記番号:新540

A Structural Approach to Tree Decompositions of Knots and Spatial Graphs

Knots are commonly represented and manipulated via diagrams, which are decorated planar graphs. When such a knot diagram has low treewidth, parameterized graph algorithms can be leveraged to ensure the fast computation of many invariants and properties of the knot. It was recently proved that there exist knots which do not admit any diagram of low treewidth, and the proof relied on intricate low-dimensional topology techniques. In this work, we initiate a thorough investigation of tree decompositions of knot diagrams (or more generally, diagrams of spatial graphs) using ideas from structural graph theory. We define an obstruction on spatial embeddings that forbids low tree width diagrams, and we prove that it is optimal with respect to a related width invariant. We then show the existence of this obstruction for knots of high representativity, which include for example torus knots, providing a new and self-contained proof that those do not admit diagrams of low treewidth. This last step is inspired by a result of Pardon on knot distortion

Physically Interacting With Four Dimensions

Thesis (Ph.D.) - Indiana University, Computer Sciences, 2009People have long been fascinated with understanding the fourth dimension. While making pictures of 4D objects by projecting them to 3D can help reveal basic geometric features, 3D graphics images by themselves are of limited value. For example, just as 2D shadows of 3D curves may have lines crossing one another in the shadow, 3D graphics projections of smooth 4D topological surfaces can be interrupted where one surface intersects another. The research presented here creates physically realistic models for simple interactions with objects and materials in a virtual 4D world. We provide methods for the construction, multimodal exploration, and interactive manipulation of a wide variety of 4D objects. One basic achievement of this research is to exploit the free motion of a computer-based haptic probe to support a continuous motion that follows the \emph{local continuity\/} of a 4D surface, allowing collision-free exploration in the 3D projection. In 3D, this interactive probe follows the full local continuity of the surface as though we were in fact \emph{physically touching\/} the actual static 4D object. Our next contribution is to support dynamic 4D objects that can move, deform, and collide with other objects as well as with themselves. By combining graphics, haptics, and collision-sensing physical modeling, we can thus enhance our 4D visualization experience. Since we cannot actually place interaction devices in 4D, we develop fluid methods for interacting with a 4D object in its 3D shadow image using adapted reduced-dimension 3D tools for manipulating objects embedded in 4D. By physically modeling the correct properties of 4D surfaces, their bending forces, and their collisions in the 3D interactive or haptic controller interface, we can support full-featured physical exploration of 4D mathematical objects in a manner that is otherwise far beyond the real-world experience accessible to human beings