Narrow Results

In this paper, we construct a conformally invariant functional for two-component links called the zone modulus of the link. Its main property is to give a sufficient condition for a link to be split.

The zone modulus is a positive number, and its lower bound is 1. To construct a link with modulus arbitrarily close to 1, it is sufficient to consider two small disjoint spheres each one far from the other and then to construct a link by taking a circle enclosed in each sphere. Such a link is a split link. The situation is different when the link is non-split: we will prove that the modulus of a non-split link is greater than . This value of the modulus is realized by a special configuration of linked circles called the Clifford link.

As a corollary, we show that if the thickness of a non-split two-component link embedded in S³ is equal to , then the link is the standard geometric Hopf link.

We establish a fundamental connection between smooth and polygonal knot energies, showing that the Minimum Distance Energy for polygons inscribed in a smooth knot converges to the Möbius Energy of the smooth knot as the polygons converge to the smooth knot. For this to work, the polygons must converge in a "nice" way, and the energies must be correctly regularized. We determine an explicit error bound for the convergence.

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.

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.

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.

We investigate a discrete version of the Möbius energy, that is of geometric interest in its own right and is defined on equilateral polygons with n segments. We show that the Γ-limit regarding L^q or W^1,q convergence, q ∈ [1, ∞] of these energies as n → ∞ is the smooth Möbius energy. This result directly implies the convergence of almost minimizers of the discrete energies to minimizers of the smooth energy if we can guarantee that the limit of the discrete curves belongs to the same knot class. Additionally, we show that the unique minimizer amongst all polygons is the regular n-gon. Moreover, discrete overall minimizers converge to the round circle.

It is well known that one of O’Hara’s knot energies is called the Möbius energy because of its invariance under Möbius transformations. We showed in a previous paper that the Möbius energy can be decomposed into three parts that retain invariance but we left open the question of invariance regarding inversions with respect to spheres centered on a knot. Here, we answer this question under the assumption that the knots have extra regularity. The result holds not only for knots but also for closed curves in ${ℝ}^n$.

We discuss the infinitesimal bending of curves and knots in ${{ℛ}}^3$. A brief overview of the results on the infinitesimal bending of curves is outlined. Change of the Willmore energy, as well as of the Möbius energy under infinitesimal bending of knots is considered. Our visualization tool devoted to visual representation of infinitesimal bending of knots is presented.