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 di&#64256erent sub-disciplines, such as the young &#64257eld of geometric knot theory, combinatorial knot theory, as well as applications in microbiology and theoretical physics.

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.