The aim of this book is to present recent results in both theoretical and applied knot theory-which are at the same time stimulating for leading researchers in the αeld as well as accessible to non-experts. The book comprises recent research results while covering a wide range of di erent sub-disciplines, such as the young αeld of geometric knot theory, combinatorial knot theory, as well as applications in microbiology and theoretical physics.

• 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