Energy Of Knots And Conformal GeometryEnergy 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. |
Contents
Part 2 Energy of knots from a conformal geometric viewpoint | 133 |
Appendix A Generalization of the Gauss formula for the linking number | 243 |
Appendix B The 3tuple map to the set of circles in S3 | 257 |
Appendix C Conformal moduli of a solid torus | 259 |
Appendix D Kirchhoff elastica | 263 |
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Common terms and phrases
2-plane 2-sphere acº annulus antipodal average crossing number base circle boundary spheres C*-topology concircular conformal angle conformal geometry Conjecture contains converges counter term cross-separating annulus defined Definition denote diagonal dsdt energy minimizers equal finite formula four points geometric given hence Hopf link ILO1 implies infimum infinitesimal cross ratio integrand intersection knot energy functional knot h knot K knot theory knot type L-orthogonal Langevin LO1 Lemma length linking number Lorentz group Math measure of non-trivial Möbius invariant Möbius transformation non-trivial annulus non-trivial knot non-trivial spheres osculating circle pair of points parametrized by arc-length pencil polygonal knot positive constant prime knot Proposition remark S1 and S2 satisfies self-repulsive solid torus space-like stereographic projection subarc Suppose tangent vector Theorem time-like vector torus knot trivial knot twice tangent sphere XX(a