Related papers: Recent topics on the O'Hara energies
We considered random discrete approximation of O'Hara energy. O'Hara energy is the energy defined for a knot, and O'Hara energy was introduced for defining the standard shape for each knot class (equivalence class by ambient isotopy) by…
We develop a regularity theory for extremal knots of scale invariant knot energies defined by J. O'hara in 1991. This class contains as a special case the M\"obius energy. For the M\"obius energy, due to the celebrated work of Freedman, He,…
O'Hara's energies, introduced by Jun O'Hara, were proposed to answer the question of what is the canonical shape in a given knot type, and were configured so that the less the energy value of a knot is, the "better" its shape is. The…
The M\"{o}bius energy, defined by O'Hara, is one of the knot energies, and named after the M\"{o}bius invariant property which was shown by Freedman-He-Wang. The energy can be decomposed into three parts, each of which is M\"{o}bius…
O'Hara introduced several functionals as knot energies. One of them is the M\"{o}bius energy. We know its M\"{o}bius invariance from Doyle-Schramm's cosine formula. It is also known that the M\"{o}bius energy was decomposed into three…
In this paper, we propose a discrete version of O'Hara's knot energy defined on polygons embedded in the Euclid space. It is shown that values of the discrete energy of polygons inscribing the curve which has bounded O'Hara's energy…
Jun O'Hara invented a family of knot energies $E^{j,p}$, $j,p \in (0, \infty)$. We study the negative gradient flow of the sum of one of the energies $E^\alpha = E^{\alpha,1}$, $\alpha \in (2,3)$, and a positive multiple of the length.…
In the present paper we investigate generalizations of O'Hara's M\"obius energy on curves \cite{ohara_1991a}, to M\"obius-invariant energies on non-smooth subsets of $\R^n$ of arbitrary dimension and co-dimension. In particular, we show…
We introduce a new discretization of O'Hara's M\"obius energy. In contrast to the known discretizations of Simon and Kim and Kusner it is invariant under M\"obius transformations of the surrounding space. The starting point for this new…
We define a new class of knot energies (known as renormalization energies) and prove that a broad class of these energies are uniquely minimized by the round circle. Most of O'Hara's knot energies belong to this class. This proves two…
We generalize the notion of integral Menger curvature introduced by Gonzalez and Maddocks by decoupling the powers in the integrand. This leads to a new two-parameter family of knot energies $intM^{p,q}$. We classify finite-energy curves in…
Plateau problems with elastic boundary energies have been of recent theoretical and applied interest. However, strong assumptions have to be made to avoid self-intersections of the boundary curve during energy minimization. We introduce a…
In this short article, we extend the cosine formula for the M\"{o}bius energy to generalized O'Hara energies. The newly derived formula gives us a condition for which the right circle minimizes the energy under the length-constraint.…
We prove that if a curve parametrized by arc length is a stationary point of the Moebius energy introduced by Jun O'Hara, then it is smooth whenever the Moebius energy is finite. Our methods, interestingly, only rely on purely analytical…
A physically natural potential energy for simple closed curves in $\bold R^3$ is shown to be invariant under M\"obius transformations. This leads to the rapid resolution of several open problems: round circles are precisely the absolute…
A new type of knot energy is presented via real life experiments involving a thin resilient metallic tube. Knotted in different ways, the device mechanically acquires a uniquely determined (up to isometry) normal form at least when the…
We prove the analyticity of smooth critical points for O'Hara's knot energies $\mathcal{E}^{\alpha,p}$, with $p=1$ and $2<\alpha< 3$, subject to a fixed length constraint. This implies, together with the main result in \cite{BR13}, that…
The present chapter gives an overview on results for discrete knot energies. These discrete energies are designed to make swift numerical computations and thus open the field to computational methods. Additionally, they provide an…
Let $E_f$ be the energy of some knot $\tau$ for any $f$ from certain class of functions. The problem is to find knots with extremal values of energy. We discuss the notion of the locally perturbed knot. The knot circle minimizes some…
In this article we study the gradient flow of the M\"obius energy introduced by O'Hara in 1991. We will show a fundamental $\varepsilon$-regularity result that allows us to bound the infinity norm of all derivatives for some time if the…