Energy of a Knot

Fukuhara, Shinji

doi:10.1016/b978-0-12-480440-1.50025-3

Cited by 38 publications

(29 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In search of optimal representatives of given knot classes Fukuhara [23] proposed the concept of knot energies as functionals defined on the space of knots, providing infinite energy barriers between different knot types. This concept was made more precise later and was investigated by various authors; see e.g.…”

Section: Introductionmentioning

confidence: 99%

On some knot energies involving Menger curvature

Strzelecki

Szumańska

Mosel

2013

Topology and its Applications

View full text Add to dashboard Cite

We investigate knot-theoretic properties of geometrically defined curvature energies such as integral Menger curvature. Elementary radii-functions, such as the circumradius of three points, generate a family of knot energies guaranteeing self-avoidance and a varying degree of higher regularity of finite energy curves. All of these energies turn out to be charge, minimizable in given isotopy classes, tight and strong. Almost all distinguish between knots and unknots, and some of them can be shown to be uniquely minimized by round circles. Bounds on the stick number and the average crossing number, some non-trivial global lower bounds, and unique minimization by circles upon compaction complete the picture.Comment: 31 pages, 4 figures; version 2 with minor changes and modification

show abstract

Section: Introductionmentioning

confidence: 99%

On some knot energies involving Menger curvature

Strzelecki

Szumańska

Mosel

2013

Topology and its Applications

View full text Add to dashboard Cite

show abstract

“…The motion of a knotted charged fiber within a viscous liquid served as model for the definition of so-called knot energies introduced by Fukuhara [11]. One hopes that it will reach a stationary point minimizing its electrostatic energy and that the resulting shape will help to determine its knot type.…”

Section: Introductionmentioning

confidence: 99%

Stationary points of O’Hara’s knot energies

Blatt

Reiter

2012

manuscripta math.

View full text Add to dashboard Cite

In this article we study the regularity of stationary points of the knot energies E (α) introduced by O'Hara in [14,15,16] in the range α ∈ (2, 3). In a first step we prove that E (α) is C 1 on the set of all regular embedded curves belonging to H (α+1)/2,2 (R/Z, R n ) and calculate its derivative. After that we use the structure of the Euler-Lagrange equation to study the regularity of stationary points of E (α) plus a positive multiple of the length. We show that stationary points of finite energy are of class C ∞ -so especially all local minimizers of E (α) among curves with fixed length are smooth.

show abstract

“…We will see that for certain parameters the energy TP (p,q) is a knot energy. The notion of knot energies goes back to Fukuhara [21] and O'Hara [36]. The general idea is to search for a "nicely shaped" representative in a given knot class having strands being widely apart and being preferably smooth.…”

Section: Introductionmentioning

confidence: 99%

Regularity theory for tangent-point energies: The non-degenerate sub-critical case

Blatt

Reiter

2014

Advances in Calculus of Variations

View full text Add to dashboard Cite

In this article we introduce and investigate a new two-parameter family of knot energies TP (p,q) that contains the tangent-point energies. These energies are obtained by decoupling the exponents in the numerator and denominator of the integrand in the original definition of the tangent-point energies.We will first characterize the curves of finite energy TP (p,q) in the sub-critical range p ∈ (q + 2, 2q + 1) and see that those are all injective and regular curves in the Sobolev-Slobodeckiȋ space W (p−1)/q,q (R/Z, R n ). We derive a formula for the first variation that turns out to be a non-degenerate elliptic operator for the special case q = 2 -a fact that seems not to be the case for the original tangent-point energies. This observation allows us to prove that stationary points of TP (p,2) + λ length, p ∈ (4, 5), λ > 0, are smooth -so especially all local minimizers are smooth.

show abstract

Energy of a Knot

Cited by 38 publications

References 0 publications

On some knot energies involving Menger curvature

On some knot energies involving Menger curvature

Stationary points of O’Hara’s knot energies

Regularity theory for tangent-point energies: The non-degenerate sub-critical case

Contact Info

Product

Resources

About