Book embeddings of Reeb graphs

Let $X$ be a simplicial complex with a piecewise linear function $f:X\to\mathbb{R}$. The Reeb graph $Reeb(f,X)$ is the quotient of $X$, where we collapse each connected component of $f^{-1}(t)$ to a single point. Let the nodes of $Reeb(f,X)$ be all homologically critical points where any homology of the corresponding component of the level set $f^{-1}(t)$ changes. Then we can label every arc of $Reeb(f,X)$ with the Betti numbers $(\beta_1,\beta_2,\dots,\beta_d)$ of the corresponding $d$-dimensional component of a level set. The homology labels give more information about the original complex $X$ than the classical Reeb graph. We describe a canonical embedding of a Reeb graph into a multi-page book (a star cross a line) and give a unique linear code of this book embedding.Comment: 12 pages, 5 figures, more examples will be at http://kurlin.or

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

A Homologically Persistent Skeleton is a fast and robust descriptor of interest points in 2D images

2D images often contain irregular salient features and interest points with non-integer coordinates. Our skeletonization problem for such a noisy sparse cloud is to summarize the topology of a given 2D cloud across all scales in the form of a graph, which can be used for combining local features into a more powerful object-wide descriptor. We extend a classical Minimum Spanning Tree of a cloud to a Homologically Persistent Skeleton, which is scale-and-rotation invariant and depends only on the cloud without extra parameters. This graph (1) is computable in time O(nlogn) for any n points in the plane; (2) has the minimum total length among all graphs that span a 2D cloud at any scale and also have most persistent 1-dimensional cycles; (3) is geometrically stable for noisy samples around planar graphs

Exactly computable and continuous metrics on isometry classes of finite and 1-periodic sequences

The inevitable noise in real measurements motivates the problem to continuously quantify the similarity between rigid objects such as periodic time series and proteins given by ordered points and considered up to isometry maintaining inter-point distances. The past work produced many Hausdorff-like distances that have slow or approximate algorithms due to minimizations over infinitely many isometries. For finite and 1-periodic sequences under isometry in any high-dimensional Euclidean space, we introduce continuous metrics with faster algorithms. The key novelty in the periodic case is the continuity of new metrics under perturbations that change the minimum period.Comment: 16 pages, 6 figures. The second version includes extra examples to illustrate the key results. The latest version is at http://kurlin.org/projects/periodic-geometry-topology/metric1D.pd

A fast and robust algorithm to count topologically persistent holes in noisy clouds

Preprocessing a 2D image often produces a noisy cloud of interest points. We study the problem of counting holes in noisy clouds in the plane. The holes in a given cloud are quantified by the topological persistence of their boundary contours when the cloud is analyzed at all possible scales. We design the algorithm to count holes that are most persistent in the filtration of offsets (neighborhoods) around given points. The input is a cloud of n points in the plane without any user-defined parameters. The algorithm has a near linear time and a linear space O(n). The output is the array (number of holes, relative persistence in the filtration). We prove theoretical guarantees when the algorithm finds the correct number of holes (components in the complement) of an unknown shape approximated by a cloud

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

Relaxed Disk Packing

Motivated by biological questions, we study configurations of equal-sized disks in the Euclidean plane that neither pack nor cover. Measuring the quality by the probability that a random point lies in exactly one disk, we show that the regular hexagonal grid gives the maximum among lattice configurations.Comment: 8 pages => 5 pages of main text plus 3 pages in appendix. Submitted to CCCG 201