Random Knotting And Linking

This volume includes both rigorous asymptotic results on the inevitability of random knotting and linking, and Monte Carlo simulations of knot probability at small lengths. The statistical mechanics and topology of surfaces on the d-dimensional simple cubic lattice are investigated. The energy of knots is studied both analytically and numerically. Vassiliev invariants are investigated and used in random knot simulations. A mutation scheme which leaves the Jones polynomial unaltered is described. Applications include the investigation of RNA secondary structure using Vassiliev invariants, and the direct experimental measurement of DNA knot probability as a function of salt concentration in random cyclization experiments on linear DNA molecules. The papers in this volume reflect the diversity of interest across science and mathematics in this subject, from topology to statistical mechanics to theoretical chemistry to wet-lab molecular biology. Contents:Graph Invariants and the Topology of RNA Folding (L H Kauffman & Y Magarshak)The Functoriality of Vassiliev-Type Invariants of Links, Braids, and Knotted Graphs (T Stanford)Knotting of Regular Polygons in 3-Space (K C Millett)An Elementary Invariant of Knots (R Randell)DNA Knot Formation in Aqueous Solutions (S Y Shaw & J C Wang)Energy Functions for Polygonal Knots (J K Simon)A Statistical Study of Random Knotting Using the Vassiliev Invariants (T Deguchi & K Tsurusaki)Random Knots and Energy: Elementary Considerations (G R Buck)Statistical Mechanics and Topology of Surfaces in Zd (E J Janse van Rensburg)Unsplittability of Random Links (Y A Diao)Twist Sequences and Vassiliev Invariants (R Trapp)Global Mutation of Knots (D Rolfsen)On Random Knots (Y A Diao et al.) Readership: Mathematicians and mathematical physicists. keywords:Knots;Links;Polygonal Knots;Invariants;DNA;RNA;Energy Functions;Statistical Knot Theory;Random Knots;Mutation;Statistical Mechanics;Topology of Surfaces

The properties of knotted and linked configurations in space have long been of interest to physicists and mathematicians. More recently and more widely, they have become important to biologists, chemists, computer scientists, and engineers. The depth and breadth of their applications are widely appreciated. Nevertheless, fundamental and challenging questions remain to be answered. Based on a Special Session at the AMS Sectional Meeting in Las Vegas (NV) in April 2001, this volumediscusses critical questions and introduces new ideas that will stimulate multi-disciplinary applications. Some of the papers are primarily theoretical; others are experimental. Some are purely mathematical; others deal with applications of mathematics to theoretical computer science, engineering,physics, biology, or chemistry. Connections are made between classical knot theory and the physical world of macromolecules, such as DNA, geometric linkages, rope, and even cooked spaghetti. This book introduces the world of physical knot theory in all its manifestations and points the way for new research. It is suitable for a diverse audience of mathematicians, computer scientists, engineers, biologists, chemists, and physicists.

This volume consists of ten lectures given at an international workshop/conference on knot theory held in July 1996 at Waseda University Conference Center. It was organised by the International Research Institute of Mathematical Society of Japan. The workshop was attended by nearly 170 mathematicians from Japan and 14 other countries, most of whom were specialists in knot theory. The lectures can serve as an introduction to the field for advanced undergraduates, graduates and also researchers working in areas such as theoretical physics. Contents:Tunnel Number and Connected Sum of Knots (K Morimoto)Topological Imitations (A Kawauchi)Surfaces in 4-Space: A View of Normal Forms and Braidings (S Kamada)Knot Types of Satellite Knots and Twisted Knots (K Motegi)Random Knots and Links and Applications to Polymer Physics (T Deguchi & K Tsurusaki)Knots and Diagrams (L H Kauffman)On Spatial Graphs (K Taniyama)Energy and Length of Knots (G Buck & J Simon)Chern-Simons Perturbative Invariants (T Kohno)Combinatorial Methods in Dehn Surgery (C M Gordon) Readership: Mathematicians and mathematical physicists. keywords:Lectures;Knots;Conference;Proceedings;Tokyo (Japan)

This volume contains the proceedings of the AMS Special Session on Algebraic and Combinatorial Structures in Knot Theory and the AMS Special Session on Spatial Graphs, both held from October 24–25, 2015, at California State University, Fullerton, CA. Included in this volume are articles that draw on techniques from geometry and algebra to address topological problems about knot theory and spatial graph theory, and their combinatorial generalizations to equivalence classes of diagrams that are preserved under a set of Reidemeister-type moves. The interconnections of these areas and their connections within the broader field of topology are illustrated by articles about knots and links in spatial graphs and symmetries of spatial graphs in and other 3-manifolds.

Energy of knots is a theory that was introduced to create a “canonical configuration” of a knot — a beautiful knot which represents its knot type. This book introduces several kinds of energies, and studies the problem of whether or not there is a “canonical configuration” of a knot in each knot type. It also considers this problems in the context of conformal geometry. The energies presented in the book are defined geometrically. They measure the complexity of embeddings and have applications to physical knotting and unknotting through numerical experiments. Contents:In Search of the “Optimal Embedding” of a Knot:α-Energy Functional E(α)On E(2)Lp Norm Energy with Higher IndexNumerical ExperimentsStereo Pictures of E(2)MinimizersEnergy of Knots in a Riemannian ManifoldPhysical Knot EnergiesEnergy of Knots from a Conformal Geometric Viewpoint:Preparation from Conformal GeometryThe Space of Non-Trivial Spheres of a KnotThe Infinitesimal Cross RatioThe Conformal Sin Energy Esin θMeasure of Non-Trivial SpheresAppendices:Generalization of the Gauss Formula for the Linking NumberThe 3-Tuple Map to the Set of Circles in S3Conformal Moduli of a Solid TorusKirchhoff ElasticaOpen Problems and Dreams Readership: Graduate students and researchers in geometry & topology and numerical & computational mathematics. Keywords:Energy of Knots;Optimal Embedding;Conformal Geometry;Möbius Transformation;Infinitesimal Cross RatioReviews:“… this book gives an excellent, state of the art account of many of the important results in this field. A strong point of the book is that almost all proofs of the theorems stated are included.”Zentralblatt MATH

Based on a Special Session at the AMS Sectional Meeting in Las Vegas (NV) in April 2001, this volume discusses critical questions and new ideas in the areas of knotting and folding of curves in surfaces in three-dimensional space and applications of these ideas to biology, chemistry, computer science, and engineering. Some of the papers are primarily theoretical; others are experimental. Some are purely mathematical; others deal with applications of mathematics to theoretical computer science, engineering, physics, biology, or chemistry. Connections are made between classical knot theory and the physical world of macromolecules, such as DNA, geometric linkages, rope, and even cooked spaghetti. This book introduces the world of physical knot theory in all its manifestations and points the way for new research. It is suitable for a diverse audience of mathematicians, computer scientists, engineers, biologists, chemists, and physicists.

This book contains the proceedings of two international conferences: a satellite meeting of the IUPAP Statphys-19 Conference and the Seventh Nankai Workshop, held in Tianjin, China in August 1995. The central theme of the two conferences, which drew participants from 18 countries, was the Yang–Baxter equation and its development and applications. With topics ranging from quantum groups, vertex and spin models, to applications in condensed matter physics, this book reflects the current research interest of integrable systems in statistical mechanics. Contents:Satellite Meeting of Statphys-19:Boundary Yang–Baxter in the RSOS/SOS Representation (C Ahn & W M Koo)Quantum Domains in Ferromagnetic Anisotropic Heisenberg Chains (F C Alcaraz et al.)The Generalized Chiral Clock Model and Its Phase Diagram (H Au-Yang & J H H Perk)Algebraic Solution of the Coincidence Problem for Crystals and Quasicrystals (M Baake)Reflection Equations and Surface Critical Phenomena (M T Batchelor)Quantum Field Theories in Terms of Group-Valued Local Fields: An Overview (L-L Chau)U(1)-Invariant Local and Integrable Lattice Formulation of the Massive Thirring Model (C Destri)Dilute Algebras and Solvable Lattice Models (U Grimm)Mutual Exclusion Statistics in the Exactly Solvable Model of the Mott Metal-Insulator Transition (Y Hatsugai et al.) Quantum Group and the Hofstadter Problem (Y Hatsugai et al.)Domain Walls in the Spin-S Quantum Ising Chain (M Henkel)Probability of Phase Separation and Two Point Temperature Correlation Functions for the Bose Gas with Delta Interaction (A R Its & V E Korepin)Stochastic Reaction-Diffusion Processes, Operator Algebras and Integrable Quantum Spin Chains (G M Schütz)Vertex-Face Correspondence in Elliptic Solutions of the Yang–Baxter Equation (Y Shibukawa)Logarithmic Anomalies of Susceptibility for Solvable Models (M Takahashi)On Chiral Hubbard Model at Strong Interaction (D F Wang)Soluble Free-Fermion Models in d Dimensions (F Y Wu)Bosonization Based on Bethe Ansatz Equations and Proof of the Conformal Conjecture (Y-S Wu & Y Yu)and other papersThe Seventh Nankai Workshop:Corner Transfer Matrix of Asymmetric Vertex Models (H-P Eckle)Scaling Properties of the Ising Model in a Field (U Grimm & B Nienhuis)One Dimensional Lattice Models of Electrons with r–2 Hopping and Exchange (Ch Gruber & D F Wang)Symmetry Group Invariants for Spontaneous Magnetization (J-M Maillard)Experimental Realizations of Integrable Reaction-Diffusion Processes in Biological and Chemical Systems (G M Schütz)Zamolodchikov–Faddeev Algebra in 2-Component Anyons (Y-L Shen & M-L Ge)and other papers Readership: Theoretical physicists and mathematicians. keywords:

Knots are familiar objects. Yet the mathematical theory of knots quickly leads to deep results in topology and geometry. This work offers an introduction to this theory, starting with our understanding of knots. It presents the applications of knot theory to modern chemistry, biology and physics.