New Directions in Geometric and Applied Knot Theory

9783110571493
9783110571493
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