1.

Record Nr.

UNINA9910454278703321

Autore

O'Hara Jun

Titolo

Energy of knots and conformal geometry [[electronic resource] /] / Jun O'Hara

Pubbl/distr/stampa

River Edge, NJ, : World Scientific, c2003

ISBN

1-281-93571-9

9786611935719

981-279-530-8

Descrizione fisica

1 online resource (306 p.)

Collana

K & E series on knots and everything ; ; v. 33

Disciplina

514.224

514/.224

Soggetti

Knot theory

Conformal geometry

Electronic books.

Lingua di pubblicazione

Inglese

Formato

Materiale a stampa

Livello bibliografico

Monografia

Note generali

Description based upon print version of record.

Nota di bibliografia

Includes bibliographical references (p. 271-284) and index.

Nota di contenuto

Contents               ; Preface              ; Part 1 In search of the ""optimal embedding"" of a knot                                                            ; Chapter 1 Introduction                             ; 1.1 Motivational problem                               ; 1.2 Notations and remarks                                ; Chapter 2 a-energy functional E(a)                                         ; 2.1 Renormalizations of electrostatic energy of charged knots

2.2 Renormalizations of r-a-modified electrostatic energy Ea                                                                   2.3 Asymptotic behavior of r-a energy of polygonal knots                                                               ; 2.4 The self-repulsiveness of E( a )                                           ; Chapter 3 On E(2)                        ; 3.1 Continuity                     ; 3.2 Behavior of E(2) under ""pull-tight""                                              ; 3.3 Mobius invariance

3.4 The cosine formula for E(2)                                      3.5 Existence of E(2) minimizers                                       ; 3.6 Average crossing number and finiteness of knot types                                                               ; 3.7 Gradient regularity of E(2) minimizers and criterion of criticality                                                                              ; 3.8 Unstable E(2)-critical torus knots                                             ; 3.9 Energy associated to a diagram


3.9.1 General framework                              3.9.2 ""X-energy""                       ; 3.10 Normal projection energies                                      ; 3.11 Generalization to higher dimensions                                               ; Chapter 4 Lp norm energy with higher index                                                 ; 4.1 Definition of (a p)-energy functional for knots eap                                                              ; 4.2 Control of knots by Eap (eap)

4.3 Complete system of admissible solid tori and finiteness of knot types                                                                                4.4 Existence of Eap minimizers                                      ; 4.5 The circles minimize Eap                                   ; 4.6 Definition of a-energy polynomial for knots                                                      ; 4.7 Brylinski's beta function for knots                                              ; 4.8 Other Lp-norm energies

Chapter 5 Numerical experiments

Sommario/riassunto

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. <br><i>Contents:</i><ul><li><i><i