## Energy of Knots and Conformal GeometryEnergy of knots is a theory that was introduced to create a OC canonical configurationOCO of a knot OCo 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 OC canonical configurationOCO 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. Contents: In Search of the OC Optimal EmbeddingOCO of a Knot: -Energy Functional E (); 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 (c) 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 S 3; Conformal Moduli of a Solid Torus; Kirchhoff Elastica; Open Problems and Dreams. Readership: Graduate students and researchers in geometry & topology and numerical & computational mathematics." |

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### Contents

Introduction | 3 |

On E2 | 31 |

L norm energy with higher index | 63 |

Numerical experiments | 83 |

Stereo pictures of E2 minimi2ers | 91 |

Energy of knots in a Riemannian manifold | 101 |

Physical knot energies | 117 |

Preparation from conformal geometry | 135 |

The conformal sin energy Ens | 213 |

Measure of nontrivial spheres | 223 |

Appendix A Generali2ation of the Gauss formula for the link | 243 |

Average crossing number | 249 |

Appendix B The 3tuple map to the set of circles in S2 | 257 |

Kirchhoff elastica | 263 |

271 | |

Index | 285 |

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### Common terms and phrases

2-plane 2-sphere 4-tuple map ambient space antipodal average crossing number base circle boundary spheres Clifford torus conformal angle conformal geometry conformal transportation Conjecture converges counter term defined Definition denote diagonal dsdt electrostatic energy energy minimi2ers equal finite formula four points geodesic given hence Hopf fibration Hopf link implies infimum infinitesimal cross ratio integrand intersection inversion inverted open knot knot energy functional knot h knot type L-orthogonal Langevin LOl Lemma length light cone light-like linking number Lorent2 maximal cross-separating annulus measure of non-trivial Mobius invariant Mobius transformation non-trivial annulus non-trivial knot non-trivial spheres oriented 2-sphere osculating circle pair of points parametri2ed by arc-length polygonal knot positive constant prime knot Proposition pull-tight radius of curvature remark satisfies self-repulsive solid torus space-like stereographic projection subarc subspace Suppose tangent vector Theorem time-like vector torus knot trivial knot twice tangent sphere unit tangent vector