Untangling polygons and graphs

Untangling is a process in which some vertices of a planar graph are moved to obtain a straight-line plane drawing. The aim is to move as few vertices as possible. We present an algorithm that untangles the cycle graph C_n while keeping at least \Omega(n^{2/3}) vertices fixed. For any graph G, we also present an upper bound on the number of fixed vertices in the worst case. The bound is a function of the number of vertices, maximum degree and diameter of G. One of its consequences is the upper bound O((n log n)^{2/3}) for all 3-vertex-connected planar graphs.Comment: 11 pages, 3 figure

A polynomial bound for untangling geometric planar graphs

To untangle a geometric graph means to move some of the vertices so that the resulting geometric graph has no crossings. Pach and Tardos [Discrete Comput. Geom., 2002] asked if every n-vertex geometric planar graph can be untangled while keeping at least n^\epsilon vertices fixed. We answer this question in the affirmative with \epsilon=1/4. The previous best known bound was \Omega((\log n / \log\log n)^{1/2}). We also consider untangling geometric trees. It is known that every n-vertex geometric tree can be untangled while keeping at least (n/3)^{1/2} vertices fixed, while the best upper bound was O(n\log n)^{2/3}. We answer a question of Spillner and Wolff [arXiv:0709.0170 2007] by closing this gap for untangling trees. In particular, we show that for infinitely many values of n, there is an n-vertex geometric tree that cannot be untangled while keeping more than 3(n^{1/2}-1) vertices fixed. Moreover, we improve the lower bound to (n/2)^{1/2}.Comment: 14 pages, 7 figure

Critical exponents for random knots

The size of a zero thickness (no excluded volume) polymer ring is shown to scale with chain length $N$ in the same way as the size of the excluded volume (self-avoiding) linear polymer, as $N^{\nu}$, where $\nu \approx 0.588$. The consequences of that fact are examined, including sizes of trivial and non-trivial knots.Comment: 4 pages, 0 figure

A Transfer Matrix Approach to Studying the Entanglement Complexity of Self-Avoiding Polygons in Lattice Tubes

Self-avoiding polygons (SAPs) are a well-established useful model of ring polymers and they have also proved useful for addressing DNA topology questions. Motivated by exploring the effects of confinement on DNA topology, in this thesis, SAPs are confined to a tubular sublattice of the simple cubic lattice. Transfer matrix methods are applied to examine the entanglement complexity of SAPs in lattice tubes. Transfer matrices are generated for small tube sizes, and exact enumeration of knotting distributions are obtained for small SAP sizes. Also, a novel sampling procedure that utilizes the generated transfer matrices is implemented to obtain independent uniformly distributed random samples of large SAPs in tubes. Using these randomly generated polygons, asymptotic growth rates for the number of fixed knot-type SAPs are estimated, and evidence is provided to support a conjectured asymptotic form for the growth of the number of fixed knot-type polygons of a given size. In particular, the evidence supports that the entropic critical exponent goes up by one with each knot factor. Additionally, a system consisting of two SAPs (called a 2SAP) in a tube is also studied to explore linking. New transfer matrices are generated for 2SAPs in small tube sizes, and exact enumeration of linking distributions are obtained for small 2SAP sizes. A sampling procedure similar to that developed for SAPs is implemented by using the 2SAP transfer matrices to obtain independent uniform samples of large 2SAPs in tubes. An asymptotic form for the number of fixed link-type 2SAPs is conjectured with some supporting evidence from the sampled 2SAPs. All the evidence obtained supports the conclusion that the knotted parts in long polymers confined to tubular environments occur in a relatively localized manner. This is supported by the entropic critical exponent results, and by preliminary evidence that average spans of knot factor patterns are not growing significantly with polygon size. Similar evidence is obtained for the knotted parts in 2SAPs. The SAP study has also revealed further characteristics of knotting in tubes. For example, when the cross-sectional area of tubes are equal, evidence indicates that knotting is more likely in more symmetrical tubes as opposed to flatter tubes. Additionally, two types of knot pattern modes have been observed and strong evidence is provided that the so-called non-local mode is dominant for small tube sizes. These two modes have also been observed in non-equilibrium simulations and in DNA nanopore experiments. The evidence for the characteristics of the linked part of 2SAPs in a tube is less conclusive but its study has opened up numerous interesting questions for further study. In summary, the novelty of the contributions in the thesis include both computational and polymer modelling contributions. Computationally: transfer matrices, Monte Carlo methods, and a novel approach for knot identification for knots in tubes are developed and extended to larger tube sizes than ever before. Polymer modelling: strong numerical evidence supporting knot localization for polymers in tubes and the first evidence regarding characterising linking for polymers in tubes are obtained

On the Obfuscation Complexity of Planar Graphs

Being motivated by John Tantalo's Planarity Game, we consider straight line plane drawings of a planar graph $G$ with edge crossings and wonder how obfuscated such drawings can be. We define $obf(G)$, the obfuscation complexity of $G$, to be the maximum number of edge crossings in a drawing of $G$. Relating $obf(G)$ to the distribution of vertex degrees in $G$, we show an efficient way of constructing a drawing of $G$ with at least $obf(G)/3$ edge crossings. We prove bounds (\delta(G)^2/24-o(1))n^2 < \obf G <3 n^2 for an $n$-vertex planar graph $G$ with minimum vertex degree $\delta(G)\ge 2$. The shift complexity of $G$, denoted by $shift(G)$, is the minimum number of vertex shifts sufficient to eliminate all edge crossings in an arbitrarily obfuscated drawing of $G$ (after shifting a vertex, all incident edges are supposed to be redrawn correspondingly). If $\delta(G)\ge 3$, then $shift(G)$ is linear in the number of vertices due to the known fact that the matching number of $G$ is linear. However, in the case $\delta(G)\ge2$ we notice that $shift(G)$ can be linear even if the matching number is bounded. As for computational complexity, we show that, given a drawing $D$ of a planar graph, it is NP-hard to find an optimum sequence of shifts making $D$ crossing-free.Comment: 12 pages, 1 figure. The proof of Theorem 3 is simplified. An overview of a related work is adde

Applications of Graph Embedding in Mesh Untangling

The subject of this thesis is mesh untangling through graph embedding, a method of laying out graphs on a planar surface, using an algorithm based on the work of Fruchterman and Reingold[1]. Meshes are a variety of graph used to represent surfaces with a wide number of applications, particularly in simulation and modelling. In the process of simulation, simulated forces can tangle the mesh through deformation and stress. The goal of this thesis was to create a tool to untangle structured meshes of complicated shapes and surfaces, including meshes with holes or concave sides. The goals of graph embedding, such as minimizing edge crossings align very well with the objectives of mesh untangling. I have designed and tested a tool which I named MUT (Mesh Untangling Tool) on meshes of various types including triangular, polygonal, and hybrid meshes. Previous methods of mesh untangling have largely been numeric or optimizationbased. Additionally, most untangling methods produce low quality graphs which must be smoothed separately to produce good meshes. Currently graph embedding techniques have only been used for smoothing of untangled meshes. I have developed a tool based on the Fruchterman-Reingold algorithm for force-directed layout[1] that effectively untangles and smooths meshes simultaneously using graph embedding techniques. It can untangle complicated meshes with irregular polygonal frames, internal holes, and other complications that previous methods struggle with. The MUT does this by using several different approaches: untangling the mesh in stages from the frame in and anchoring the mesh at corner points to stabilize the untangling

Knots and Seifert surfaces

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2020, Director: Javier J. Gutiérrez Marín[en] Since the beginning of the degree that I think that everyone should have the oportunity to know mathematics as they are and not as they are presented (or were presented) at a high school level. In my opinion, the answer to the question "why do we do this?" that a student asks, shouldn’t be "because is useful", it should be "because it’s interesting" or "because we are curious". To study mathematics (in every level) should be like solving an enormous puzzle. It should be a playful experience and satisfactory (which doesn’t mean effortless nor without dedication). It is this idea that brought me to choose knot theory as the main focus of my project. I wanted a theme that generated me curiosity and that it could be attractive to other people with less mathematical background, in order to spread what mathematics are to me. It is because of this that i have dedicated quite some time to explain the intuitive idea behind every proof and definition, and it is because of this that the great majority of proofs and definitions are paired up with an image (created by me). In regards to the technical part of the project I have had as main objectives: to introduce myself to knot theory, to comprehend the idea of genus of a knot and know the propeties we could derive to study knots