Repulsive knot energies and pseudodifferential calculus for O’Hara’s knot energy family E (α) , α ∈ [2, 3)

2012 ◽  
Vol 285 (7) ◽  
pp. 889-913 ◽  
Author(s):  
Philipp Reiter
Keyword(s):  
Pseudodifferential Calculus ◽  
Knot Energy

scholarly journals TANGENT-POINT SELF-AVOIDANCE ENERGIES FOR CURVES

2012 ◽  
Vol 21 (05) ◽  
pp. 1250044 ◽  
Author(s):  
PAWEŁ STRZELECKI ◽  
HEIKO VON DER MOSEL
Keyword(s):  
Upper Bound ◽  
Closed Interval ◽  
Local Energy ◽  
Triple Junctions ◽  
Tangent Point ◽  
Double Integral ◽  
Energy Formula ◽  
Knot Energy ◽  
Rectifiable Curves ◽  
Infinite Energy

We study a two-point self-avoidance energy [Formula: see text] which is defined for all rectifiable curves in ℝn as the double integral along the curve of 1/rq. Here r stands for the radius of the (smallest) circle that the is tangent to the curve at one point and passes through another point on the curve, with obvious natural modifications of this definition in the exceptional, non-generic cases. It turns out that finiteness of [Formula: see text] for q ≥ 2 guarantees that γ has no self-intersections or triple junctions and therefore must be homeomorphic to the unit circle 𝕊1 or to a closed interval I. For q > 2 the energy [Formula: see text] evaluated on curves in ℝ3 turns out to be a knot energy separating different knot types by infinite energy barriers and bounding the number of knot types below a given energy value. We also establish an explicit upper bound on the Hausdorff-distance of two curves in ℝ3 with finite [Formula: see text]-energy that guarantees that these curves are ambient isotopic. This bound depends only on q and the energy values of the curves. Moreover, for all q that are larger than the critical exponent q crit = 2, the arclength parametrization of γ is of class C1, 1-2/q, with Hölder norm of the unit tangent depending only on q, the length of γ, and the local energy. The exponent 1 - 2/q is optimal.

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

2008 ◽  
Vol 464 (2098) ◽  
pp. 2741-2757 ◽  
Author(s):  
Fanghua Lin ◽  
Yisong Yang
Keyword(s):  
Fine Structure ◽  
Field Configuration ◽  
Hopf Fibration ◽  
Essential Characteristic ◽  
Knot Energy

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 E N stratified at the Hopf charge N of the knot energy of the Faddeev type induced from the Hopf fibration ( n ≥1) in general dimensions obeys the sharp fractional-exponent growth law , where the exponent p is universally rendered as , 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.

Méthode BKW formelle et spectre des molécules polyatomiques dans l'approximation de Born–Oppenheimer

2001 ◽  
Vol 79 (4) ◽  
pp. 757-771 ◽  
Author(s):  
B Messirdi ◽  
A Senoussaoui
Keyword(s):  
First Eigenvalue ◽  
The First Eigenvalue ◽  
Pseudodifferential Calculus ◽  
Coulomb Type ◽  
Formal Version

We studied the spectrum of P = -h2Δx – Δy + V (x,y) on L2(IRx3n × IRy3p), when h tends to zero, n [Formula: see text] 2, p [Formula: see text] IN*, in the case where the potential V(x,y) is singular and of Coulomb type and the first eigenvalue of Q (x) = -Δy + V(x,y) on L (IRy3p) admits an unbounded well. Using a formal version of the h-pseudodifferential calculus on the regularized operator of P with adapted changes, we obtained WKB-type expansions of eigenvalues and associated eigenfunctions of P.

Menger curvature as a knot energy

2013 ◽  
Vol 530 (3) ◽  
pp. 257-290 ◽  
Author(s):  
Paweł Strzelecki ◽  
Heiko von der Mosel
Keyword(s):  
Menger Curvature ◽  
Knot Energy

OPTIMALLY IMMERSED PLANAR CURVES UNDER MÖBIUS ENERGY

2011 ◽  
Vol 20 (10) ◽  
pp. 1381-1390
Author(s):  
RYAN P. DUNNING
Keyword(s):  
Calculus Of Variations ◽  
Direct Method ◽  
Energy Parameter ◽  
Energy Estimates ◽  
Singular Behavior ◽  
Planar Curves ◽  
Intersecting Curve ◽  
Knot Energy ◽  
Parameter Dependent ◽  
Selection Of

This paper investigates the existence of optimally immersed planar self-intersecting curves. Because any self-intersecting curve will have infinite knot energy, parameter-dependent renormalizations of the Möbius energy remove the singular behavior of the curve. The direct method of the calculus of variations allows for the selection of optimal immersions in various restricted classes of curves. Careful energy estimates allow subconvergence of these optimal curves as restrictions are relaxed.

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