Static knot energy, Hopf charge, and universal growth law

2006 ◽  
Vol 747 (3) ◽  
pp. 455-463 ◽  
Author(s):  
Fanghua Lin ◽  
Yisong Yang
Keyword(s):  
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.

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.

scholarly journals M\"obius invariance of knot energy

1993 ◽  
Vol 28 (1) ◽  
pp. 99-104 ◽  
Author(s):  
Steve Bryson ◽  
Michael H. Freedman ◽  
Zheng-Xu He ◽  
Zhenghan Wang
Keyword(s):  
Knot Energy

scholarly journals Knot Energy, Complexity, and Mobility of Knotted Polymers

2017 ◽  
Vol 7 (1) ◽  
Author(s):  
Fernando Vargas–Lara ◽  
Ahmed M. Hassan ◽  
Marc L. Mansfield ◽  
Jack F. Douglas
Keyword(s):  
Knot Energy

In Search of a Good Polygonal Knot Energy

1997 ◽  
Vol 06 (05) ◽  
pp. 633-657 ◽  
Author(s):  
Y. Diao ◽  
C. Ernst ◽  
E. J. Janse van Rensburg
Keyword(s):  
Energy Function ◽  
Invariant Function ◽  
Science Literature ◽  
Scale Invariant ◽  
Energy Functions ◽  
Limiting Behavior ◽  
New Energy ◽  
Knot Energy ◽  
Three Space ◽  
Polygonal Knots

An energy function on knots is a scale-invariant function from knot conformations into non-negative real numbers. The infimum of an energy function is an invariant which defines "canonical conformation(s)" of a knot in three space. These are not necessarily unique, and, in some cases, may even be singular. Many hierarchies of energy functions for knots in the mathematical and physical science literature have been studied, each energy function with its own characteristic set of properties. In this paper we focus on the energy functions of equilateral polygonal knots. These energy functions are important in computer studies of knot energies, and are often defined as discrete versions of energy functions defined on smooth knots. Energy functions on equilateral polygonal knots turn out to be ill-behaved in many cases. To characterize a "good" polygonal knot energy we introduce the concepts of asymptotically finite and asymptotically smooth energy functions of equilateral polygonal knots. Energy functions which are both asymptotically finite and smooth tend to have food limiting behavior (as the number of edges goes to infinity). We introduce a new energy function of equilateral polygonal knots, and show that it is both asymptotically finite and smooth. In addition, we compute this energy for several knots using simulated annealing.

scholarly journals DOES FINITE KNOT ENERGY LEAD TO DIFFERENTIABILITY?

2008 ◽  
Vol 17 (10) ◽  
pp. 1281-1310 ◽  
Author(s):  
SIMON BLATT ◽  
PHILIPP REITER
Keyword(s):  
Finite Energy ◽  
Complete Picture ◽  
Hölder Exponent ◽  
Hölder Continuous ◽  
First Case ◽  
Regularizing Effects ◽  
Knot Energy

In this article, we raise the question if curves of finite (j, p)-knot energy introduced by O'Hara are at least pointwise differentiable. If we exclude the highly singular range (j - 2)p ≥ 1, the answer is no for jp ≤ 2 and yes for jp > 2. In the first case, which also contains the most prominent example of the Möbius energy(j = 2, p = 1) investigated by Freedman, He and Wang, we construct counterexamples. For jp > 2, we prove that finite-energy curves have in fact a Hölder continuous tangent with Hölder exponent ½(jp - 2)/(p + 2). Thus, we obtain a complete picture as to what extent the (j, p)-energy has self-avoidance and regularizing effects for (j, p) ∈ (0, ∞) × (0, ∞). We provide results for both closed and open curves.

ERROR ANALYSIS OF THE MINIMUM DISTANCE ENERGY OF A POLYGONAL KNOT AND THE MÖBIUS ENERGY OF AN APPROXIMATING CURVE

2010 ◽  
Vol 19 (08) ◽  
pp. 975-1000 ◽  
Author(s):  
ERIC J. RAWDON ◽  
JOSEPH WORTHINGTON
Keyword(s):  
Error Analysis ◽  
Upper Bound ◽  
Minimum Distance ◽  
Energy Functionals ◽  
Explicit Bound ◽  
Knot Energy ◽  
The Given

Energy minimizing smooth knot configurations have long been approximated by finding knotted polygons that minimize discretized versions of the given energy. However, for most knot energy functionals, the question remains open on whether the minimum polygonal energies are "close" to the minimum smooth energies. In this paper, we determine an explicit bound between the Minimum-Distance Energy of a polygon and the Möbius Energy of a piecewise-C2 knot inscribed in the polygon. This bound is written in terms of the ropelength and the number of edges and can be used to determine an upper bound for the minimum Möbius Energy for different knot types.

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