Universal growth law for knot energy of Faddeev type in general dimensions

Research output: Contribution to journalArticle

Abstract

The presence of a fractional-exponent growth law relating knot energy and knot topology is known to be an essential characteristic for the existence of 'ideal' knots. In this paper, we show that the energy infimum EN stratified at the Hopf charge N of the knot energy of the Faddeev type induced from the Hopf fibration S<sup>4n-1</ sup>→S<sup>2n</sup> (n≥1) in general dimensions obeys the sharp fractional-exponent growth law EN∼|N|<sup>p</sup>, where the exponent p is universally rendered as p=(4n-1)/4n, which is independent of the detailed fine structure of the knot energy but determined completely by the dimensions of the domain and range spaces of the field configuration maps.

Original languageEnglish (US)
Pages (from-to)2741-2757
Number of pages17
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume464
Issue number2098
DOIs
StatePublished - Oct 8 2008

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Knot
exponents
Energy
Exponent
Topology
Fractional
energy
Hopf Fibration
Fine Structure
topology
fine structure
Charge
configurations
Configuration
Range of data

Keywords

  • Energy-topology growth laws
  • Hopf fibration
  • Hopf invariant
  • Ideal knots
  • Knot energy
  • Sobolev inequalities

ASJC Scopus subject areas

  • General

Cite this

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title = "Universal growth law for knot energy of Faddeev type in general dimensions",
abstract = "The presence of a fractional-exponent growth law relating knot energy and knot topology is known to be an essential characteristic for the existence of 'ideal' knots. In this paper, we show that the energy infimum EN stratified at the Hopf charge N of the knot energy of the Faddeev type induced from the Hopf fibration S4n-1→S2n (n≥1) in general dimensions obeys the sharp fractional-exponent growth law EN∼|N|p, where the exponent p is universally rendered as p=(4n-1)/4n, which is independent of the detailed fine structure of the knot energy but determined completely by the dimensions of the domain and range spaces of the field configuration maps.",
keywords = "Energy-topology growth laws, Hopf fibration, Hopf invariant, Ideal knots, Knot energy, Sobolev inequalities",
author = "Fanghua Lin and Yisong Yang",
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AU - Yang, Yisong

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AB - The presence of a fractional-exponent growth law relating knot energy and knot topology is known to be an essential characteristic for the existence of 'ideal' knots. In this paper, we show that the energy infimum EN stratified at the Hopf charge N of the knot energy of the Faddeev type induced from the Hopf fibration S4n-1→S2n (n≥1) in general dimensions obeys the sharp fractional-exponent growth law EN∼|N|p, where the exponent p is universally rendered as p=(4n-1)/4n, which is independent of the detailed fine structure of the knot energy but determined completely by the dimensions of the domain and range spaces of the field configuration maps.

KW - Energy-topology growth laws

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KW - Sobolev inequalities

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