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

Format:

Book

Author/Creator:

Blatt, Simon

Contributor:

Reiter, Philipp, editor.

Blatt, Simon, editor.

Schikorra, Armin, editor.

Language:

English

Subjects (All):

Knot theory.

Physical Description:

1 online resource (288 pages) : illustrations

Edition:

1st ed.

Place of Publication:

Berlin ; Boston : De Gruyter, [2018]

Language Note:

English

Summary:

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.

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

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

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

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

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

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

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

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

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

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

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.

Notes:

Description based on: online resource; title from PDF information screen (De Gruyter, viewed November 17, 2022).

Description based on publisher supplied metadata and other sources.

OCLC:

1105777553