arXiv subjects

Philipp Reiter

Publications and source records attributed to Philipp Reiter.

23 records · Page 2Linked to original sources

arXiv2016

The elastic trefoil is the twice covered circle

We investigate the elastic behavior of knotted loops of springy wire. To this end we minimize the classic bending energy $E_{\text{bend}}=\intκ^2$ together with a small multiple of ropelength $\mathcal R=\text{length}/\text{thickness}$ in order to penalize selfintersection. Our main objective is to characterize elastic knots, i.e., all limit configurations of energy minimizers of the total energy $E_\vartheta:=E_{\text{bend}}+\vartheta\mathcal R$ as $\vartheta$ tends to zero. The elastic unknot turns out to be the round circle with bending energy $(2π)^2$. For all (non-trivial) knot classes for which the natural lower bound $(4π)^2$ for the bending energy is sharp, the respective elastic knot is the twice covered circle. The knot classes for which $(4π)^2$ is sharp are precisely the $(2,b)$-torus knots for odd $b$ with $|b|\ge 3$ (containing the trefoil). In particular, the elastic trefoil is the twice covered circle.

math.DG↗

arXiv2013

Towards a regularity theory for integral Menger curvature

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 terms of Sobolev-Slobodeckij spaces. Moreover, restricting to the range of parameters leading to a sub-critical Euler-Lagrange equation, we prove existence of minimizers within any knot class via a uniform bi-Lipschitz bound. Consequently, $intM^{p,q}$ is a knot energy in the sense of O'Hara. Restricting to the non-degenerate sub-critical case, a suitable decomposition of the first variation allows to establish a bootstrapping argument that leads to $C^{\infty}$-smoothness of critical points.

math.AP↗

arXiv2012

Regularity theory for tangent-point energies: The non-degenerate sub-critical case

In this article we introduce and investigate a new two-parameter family of knot energies $TP^{(p,q)}$ that contains the tangent-point energies. These energies are obtained by decoupling the exponents in the numerator and denominator of the integrand in the original definition of the tangent-point energies. We will first characterize the curves of finite energy in the sub-critical range $p\in(q+2,2q+1)$ and see that those are all injective and regular curves in the Sobolev-Slobodeckiĭ space $W^{(p-1)/q,q}$. We derive a formula for the first variation that turns out to be a non-degenerate elliptic operator for the special case $q=2$ --- a fact that seems not to be the case for the original tangent-point energies. This observation allows us to prove that stationary points of $TP^{(p,2)}$ + λlength, $p\in(4,5)$, λ> 0, are smooth --- so especially all local minimizers are smooth.

math.AP↗

arXiv2012

Hard analysis meets critical knots (Stationary points of the Moebius energy are smooth)

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 arguments, entirely without using Moebius invariance. Furthermore, the techniques involved are not fundamentally restricted to one-dimensional domains, but are generalizable to arbitrary dimensions.

math.AP↗

arXiv2012

Stationary Points of O'Hara's Knot Energies

In this article we study the regularity of stationary points of the knot energies $E^α$ introduced by O'Hara in the range $α\in (2,3)$. In a first step we prove that $E^α$ is $C^1$ on the set of all regular embedded closed curves belonging to $H^{(α+1)/2,2}$ and calculate its derivative. After that we use the structure of the Euler-Lagrange equation to study the regularity of stationary points of $E^α$ plus a positive multiple of the length. We show that stationary points of finite energy are of class $C^\infty$ - so especially all local minimizers of $E^α$ among curves with fixed length are smooth.

math.AP↗