by Jun O'Hara (Tokyo Metropolitan University, Japan)

Energy of knots is a theory that was introduced to create a “canonical configuration” of a knot — a beautiful knot which represents its knot type. This book introduces several kinds of energies, and studies the problem of whether or not there is a “canonical configuration” of a knot in each knot type. It also considers this problems in the context of conformal geometry. The energies presented in the book are defined geometrically. They measure the complexity of embeddings and have applications to physical knotting and unknotting through numerical experiments.

• In Search of the “Optimal Embedding” of a Knot:
  • α-Energy Functional E^(α)
  • On E⁽²⁾
  • L^p Norm Energy with Higher Index
  • Numerical Experiments
  • Stereo Pictures of E⁽²⁾Minimizers
  • Energy of Knots in a Riemannian Manifold
  • Physical Knot Energies
• Energy of Knots from a Conformal Geometric Viewpoint:
  • Preparation from Conformal Geometry
  • The Space of Non-Trivial Spheres of a Knot
  • The Infinitesimal Cross Ratio
  • The Conformal Sin Energy E_{sin θ}
  • Measure of Non-Trivial Spheres
• Appendices:
  • Generalization of the Gauss Formula for the Linking Number
  • The 3-Tuple Map to the Set of Circles in S³
  • Conformal Moduli of a Solid Torus
  • Kirchhoff Elastica
  • Open Problems and Dreams