New directions in geometric and applied knot theory / / edited by Philipp Reiter, Simon Blatt and Armin Schikorra.

• Author: Blatt, Simon
• TeilnehmendeR: Reiter, Philipp,; Blatt, Simon,; Schikorra, Armin,
• Place / Publishing House: Berlin ;, Boston : : De Gruyter,, [2018]; ©2018
• Year of Publication: 2018
• Language: English
• Physical Description: 1 online resource (288 pages) :; illustrations

The aim of this book is to present recent results in both theoretical and applied knot theory&#8212which are at the same time stimulating for leading researchers in the &#64257eld as well as accessible to non-experts. The book comprises recent research results while covering a wide range of...

Table of Contents

• Intro
• 1 Introduction
• Geometric curvature energies: facts, trends, and open problems
• 2.1 Facts
• 2.2 Trends and open problems
• Bibliography
• On Möbius invariant decomposition of the Möbius energy
• 3.1 O'Hara's knot energies
• 3.2 Freedman-He-Wang's procedure and the Kusner-Sullivan conjecture
• 3.3 Basic properties of the Möbius energy
• 3.4 The Möbius invariant decomposition
• 3.4.1 The decomposition
• 3.4.2 Variational formulae
• 3.4.3 The Möbius invariance
• Bibliography
• Pseudogradient Flows of Geometric Energies
• 4.1 Introduction
• 4.2 Banach Bundles
• 4.2.1 General Fiber Bundles
• 4.2.2 Banach Bundles and Hilbert Bundles
• 4.3 Riesz Structures
• 4.3.1 Riesz Structures
• 4.3.2 Riesz Bundle Structures
• 4.3.3 Riesz Manifolds
• 4.4 Pseudogradient Flow
• 4.5 Applications
• 4.5.1 Minimal Surfaces
• 4.5.2 Elasticae
• 4.5.3 Euler-Bernoulli Energy and Euler Elastica
• 4.5.4 Willmore Energy
• 4.6 Final Remarks
• Bibliography
• Discrete knot energies
• 5.1 Introduction
• 5.1.1 Notation
• 5.2 Möbius Energy
• 5.3 Integral Menger Curvature
• 5.4 Thickness
• A.1 Appendix: Postlude in -convergence
• Bibliography
• Khovanov homology and torsion
• 6.1 Introduction
• 6.2 Definition and structure of Khovanov link homology
• 6.3 Torsion of Khovanov link homology
• 6.4 Homological invariants of alternating and quasi-alternating cobordisms
• Bibliography
• Quadrisecants and essential secants of knots
• 7.1 Introduction
• 7.2 Quadrisecants
• 7.2.1 Essential secants
• 7.2.2 Results about quadrisecants
• 7.2.3 Counting quadrisecants and quadrisecant approximations.
• 7.3 Key ideas in showing quadrisecants exist
• 7.3.1 Trisecants and quadrisecants.
• 7.3.2 Structure of the set of trisecants.
• 7.4 Applications of essential secants and quadrisecants
• 7.4.1 Total curvature
• 7.4.2 Second Hull.
• 7.4.3 Ropelength
• 7.4.4 Distortion
• 7.4.5 Final Remarks
• Bibliography
• Polygonal approximation of unknots by quadrisecants
• 8.1 Introduction
• 8.2 Quadrisecant approximation of knots
• 8.3 Quadrisecants of Polygonal Unknots
• 8.4 Quadrisecants of Smooth Unknots
• 8.5 Finding Quadrisecants
• 8.6 Test for Good Approximations
• Bibliography
• Open knotting
• 9.1 Introduction
• 9.2 Defining open knotting
• 9.2.1 Single closure techniques
• 9.2.2 Stochastic techniques
• 9.2.3 Other closure techniques
• 9.2.4 Topology of knotted arcs
• 9.3 Visualizing knotting in open chains using the knotting fingerprint
• 9.4 Features of knotting fingerprints, knotted cores, and crossing changes
• 9.5 Conclusions
• Bibliography
• The Knot Spectrum of Random Knot Spaces
• 10.1 Introduction
• 10.2 Basic mathematical background in knot theory
• 10.3 Spaces of random knots, knot sampling and knot identification
• 10.4 An analysis of the behavior of PK with respect to length and radius
• 10.4.1 PK(L,R) as a function of length L for fixed R
• 10.4.2 PK(L,R) as a function of confinement radius R for fixed L
• 10.4.3 Modeling PK as a function of length and radius.
• 10.5 Numerical results
• 10.5.1 The numerical analysis of PK(L,R) based on the old data
• 10.5.2 The numerical analysis of PK(L,R) based on the new data
• 10.5.3 The location of local maxima of PK(L,R)
• 10.6 The influence of the confinement radius on the distributions of knot types
• 10.6.1 3-, 4-, and 5-crossing knots
• 10.6.2 6-crossing knots
• 10.6.3 7-crossing knots
• 10.6.4 8-crossing knots
• 10.6.5 9-crossing knots
• 10.6.6 10-crossing knots
• 10.7 The influence of polygon length on the distributions of knot types in the presence of confinement
• 10.7.1 3-, 4-, and 5-crossing knots
• 10.7.2 6-crossing knots
• 10.7.3 7-crossing knots
• 10.7.4 8-crossing knots.
• 10.7.5 9-crossing knots
• 10.7.6 10-crossing knots
• 10.8 Conclusions
• Bibliography
• Sampling Spaces of Thick Polygons
• 11.1 Introduction
• 11.2 Classical Perspectives
• 11.2.1 Thickness of polygons
• 11.2.2 Self-avoiding random walks
• 11.2.3 Closed polygons: fold algorithm
• 11.2.4 Closed polygons: crankshaft algorithm
• 11.2.5 Quaternionic Perspective
• 11.3 Sampling Thick Polygons
• 11.3.1 Primer on Probability Theory
• 11.3.2 Open polygons: Plunkett algorithm ChapmanPlunkett2016
• 11.3.3 Closed polygons: Chapman algorithm
• 11.4 Discussion and Conclusions
• Bibliography
• Equilibria of elastic cable knots and links
• 12.1 Introduction
• 12.2 Theory of elastic braids made of two equidistant strands
• 12.2.1 Equidistant curves, reference frames and strains
• 12.2.2 Equations for the standard 2-braid
• 12.2.3 Kinematics equations
• 12.3 Numerical solution
• 12.3.1 Torus knots
• 12.3.2 Torus links
• 12.4 Concluding remarks
• Bibliography
• Groundstate energy spectra of knots and links: magnetic versus bending energy
• 13.1 Introduction
• 13.2 Magnetic knots and links in ideal conditions
• 13.3 The prototype problem
• 13.4 Relaxation of magnetic knots and constrained minima
• 13.5 Groundstate magnetic energy spectra
• 13.6 Bending energy spectra
• 13.7 Magnetic energy versus bending energy
• 13.8 Conclusions
• Bibliography.