TANGENT-POINT SELF-AVOIDANCE ENERGIES FOR CURVES

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.

OPTIMALLY IMMERSED PLANAR CURVES UNDER MÖBIUS ENERGY

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.

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

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.

DOES FINITE KNOT ENERGY LEAD TO DIFFERENTIABILITY?

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.

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

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.

THE MINIMUM OF KNOT ENERGY FUNCTIONS

In this paper we discuss some fundamental issues regarding knot energy functions. These include the existence of minimum values of energy functions of smooth knots and energy functions of polygonal knots within a knot type, the convergence of these minimum values in the case of polygonal knot energy and the convergence of the corresponding polygons where these minimum values are attained. When the polygonal knot energy is derived from a smooth knot energy, will the minimal polygonal knot energies converge to the infimum of the smooth knot energy? Do the corresponding polygons converge to a smooth knot at which the smooth energy achieves its minimal value? We show that one cannot expect these to be true in general and outline certain conditions that would ensure a positive answer to some of the above questions.