Sebastian Scholtes
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 independent, geometrically pleasing and consistent discrete model that behaves similarly to the original model. We will focus on Möbius energy, integral Menger curvature and thickness.
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We prove the existence of symmetric critical torus knots for O'Hara's knot energy family $E_α$, $α\in (2,3)$ using Palais' classic principle of symmetric criticality. It turns out that in every torus knot class there are at least two smooth $E_α$-critical knots, which supports experimental observations using numerical gradient flows.
We investigate knot-theoretic properties of geometrically defined curvature energies such as integral Menger curvature. Elementary radii-functions, such as the circumradius of three points, generate a family of knot energies guaranteeing self-avoidance and a varying degree of higher regularity of finite energy curves. All of these energies turn out to be charge, minimizable in given isotopy classes, tight and strong. Almost all distinguish between knots and unknots, and some of them can be shown to be uniquely minimized by round circles. Bounds on the stick number and the average crossing numb…
This paper has been withdrawn by the authors, as it was combined with "Conformally invariant energies of knots I" (math/0409396) to be "Conformally invariant energies of knots" which has replaced the former.
Circles Minimize most Knot Energies
该研究定义了一类新的重整化纽结能量，证明多数此类能量及O'Hara的多数纽结能量由正圆唯一最小化，证明了两个相关猜想，还找到非正圆最小化的能量，证明基于Luko的闭合曲线平均弦长定理。
Conformally invariant functionals on the space of knots are introduced via extrinsic conformal geometry of the knot and integral geometry on the space of spheres. Our functionals are expressed in terms of a complex-valued 2-form which can be considered as the cross-ratio of a pair of infinitesimal segments of the knot. We show that our functionals detect the unknot as the total curvature does, and that their values explode if a knot degenerates to a singular knot with double points.
On the Palais-Smale condition in geometric knot theory
本文研究了几何结理论中多种能量函数在弧长参数化结子流形上满足Palais-Smale条件，并证明了这些能量存在最小化结及临界结的光滑性。
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 Möbius energy. For the Möbius energy, due to the celebrated work of Freedman, He, and Wang, we have a relatively good understanding. Their approch is crucially based on the invariance of the Möbius energy under Möbius transforms, which fails for all the other O'hara energies. We overcome this difficulty by re-interpreting the scale invariant O'hara knot energies as a nonlinear, nonlocal $L^p$-energy acting on the unit tangent of the knot par…
We study a 1-form which can be given by a vector in a conformally invariant way. We then study conformally invariant functionals associated to a ``Y-diagram'' on the space of knots which are made from the 1-form.
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 Moebius Energy of the smooth knot as the polygons converge to the smooth knot. However, the polygons must converge in a ``nice'' way, and the energies must be correctly regularized. We determine an explicit error bound between the energies in terms of the number of the edges of the polygon and the Ropelength of the smooth curve.
We introduce and begin the study of new knot energies defined on knot diagrams. Physically, they model the internal energy of thin metallic solid tori squeezed between two parallel planes. Thus the knots considered can perform the second and third Reidemeister moves, but not the first one. The energy functionals considered are the sum of two terms, the uniformization term (which tends to make the curvature of the knot uniform) and the resistance term (which, in particular, forbids crossing changes). We define an infinite family of uniformization functionals, depending on an arbitrary smooth fu…
We prove the analyticity of smooth critical points for O'Hara's knot energies $\mathcal{E}^{α,p}$, with $p=1$ and $2<α< 3$, subject to a fixed length constraint. This implies, together with the main result in \cite{BR13}, that bounded energy critical points of $\mathcal{E}^{α,1}$ subject to a fixed length constraint are not only $C^\infty$ but also analytic. Our approach is based on Cauchy's method of majorants and a decomposition of the gradient that was adapted from the Möbius energy case $\mathcal{E}^{2,1}$ in \cite{BV19}.
We investigate minimizers and critical points for scale-invariant tangent-point energies ${\rm TP}^{p,q}$ of closed curves. We show that a) minimizing sequences in ambient isotopy classes converge to locally critical embeddings in all but finitely many points and b) show regularity of locally critical embeddings. Technically, the convergence theory a) is based on a gap-estimate of a fractional Sobolev spaces in comparison to the tangent-point energy. The regularity theory b) is based on constructing a new energy $\mathcal{E}^{p,q}$ and proving that the derivative $γ'$ of a parametrization of a…
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 s…
In this paper, we prove than given two cubic knots $K_1$, $K_2$ in $\mathbb{R}^3$, they are isotopic if and only if one can pass from one to the other by a finite sequence of cubulated moves. These moves are analogous to the Reidemeister moves for classical tame knots. We use the fact that a cubic knot is determined by a cyclic permutation of contiguous vertices of the $\mathbb{Z}^3$-lattice (with some restrictions), to describe some of the classic invariants and properties of the knots in terms of such cyclic permutations, by projecting onto a plane such that it is injective when restricted t…
We give a condition for a function to produce a Möbius invariant weighted inner product on the tangent space of the space of knots, and show that some kind of Möbius invariant knot energies can produce Möbius invariant and parametrization invariant weighted inner products. They would give a natural way to study the evolution of knots in the framework of Möbius geometry.
Using interpolation with biarc curves we prove $Γ$-convergence of discretized tangent-point energies to the continuous tangent-point energies in the $C^1$-topology, as well as to the ropelength functional. As a consequence discrete almost minimizing biarc curves converge to ropelength minimizers, and to minimizers of the continuous tangent-point energies. In addition, taking point-tangent data from a given $C^{1,1}$-curve $γ$, we establish convergence of the discrete energies evaluated on biarc curves interpolating these data, to the continuous tangent-point energy of $γ$, together with an exp…
We obtain a simple and complete characterisation of which matchings on the Tait graph of a knot diagram induce a discrete Morse function (dMf) on $S^2$, extending a construction due to Cohen. We show these dMfs are in bijection with certain rooted spanning forests in the Tait graph. We use this to count the number of such dMfs with a closed formula involving the graph Laplacian. We then simultaneously generalise Kauffman's Clock Theorem and Kenyon-Propp-Wilson's correspondence in two different directions; we first prove that the image of the correspondence induces a bijection on perfect dMfs, …
We present computational data and heuristic arguments which suggest that given a hyperbolic knot the volume correlates with its determinant, the Mahler measure of its Alexander polynomial and the Mahler measure of the twisted Alexander polynomial corresponding to the discrete and faithful SL(2,C)-representation.
Geometric considerations for energy minimization of topological links and chainmail networks
本文以少量参数刻画 Möbius 与 Minimum Distance 能量，求解 Hopf 链环、Borromean 环及锁子甲网络的最小能量几何，并给出绳长下界猜想与网络效率比较。
In this paper we use character variety methods to study homomorphisms between the fundamental groups of 3-manifolds, in particular those induced by non-zero degree maps. A {\it knot manifold} is a compact, connected, irreducible, orientable 3-manifold whose boundary is an incompressible torus. A {\it virtual epimorphism} is a homomorphism whose image is of finite index in its range. We show that the existence of such homomorphisms places constraints on the algebraic decomposition of a knot manifold's $PSL_2(\mathbb C)$-character variety and consequently determine a priori bounds on the number …
We investigate the relationship between a discrete version of thickness and its smooth counterpart. These discrete energies are defined on equilateral polygons with $n$ vertices. It will turn out that the smooth ropelength, which is the scale invariant quotient of length divided by thickness, is the $Γ$-limit of the discrete ropelength for $n\to\infty$, regarding the topology induced by the Sobolev norm $||\cdot||_{W^{1,\infty}(\mathbb{S}_{1},\mathbb{R}^{d})}$. This result directly implies the convergence of almost minimizers of the discrete energies in a fixed knot class to minimizers of the …
To a special type of grope embedded in 4-space, that we call an admissible grope, we associate a length function for each real number q at least 1. This gives rise to a family of pseudo-metrics d^q, refining the slice genus metric, on the set of concordance classes of knots, as the infimum of the length function taken over all possible grope concordances between two knots. We investigate the properties of these metrics. The main theorem is that the topology induced by this metric on the knot concordance set is not discrete for all q>1. The analogous statement for links also holds for q=1. In a…
We propose an unexpected twist to description of the geometry and topology of configurations of n straight lines considered as a whole 3D entity (because the lines are inseparably linked pairwise while having linking numbers 1/2 or -1/2) and named n-cross. Our theory stems from our work on configurations of mutually touching straight cylinders but, along with the previously introduced Ring matrix (that controls the encaging of each line by other lines), we now introduce fundamental direction 3D matrices (whose entries 0, 1, and -1 are signs of mixed products of line orientation vector triples)…
We give an algorithmic computation for the height of Kauffman's clock lattice obtained from a knot diagram with two adjacent regions starred and without crossing information specified. We show that this lattice is more familiarly the graph of perfect matchings of a bipartite graph obtained from the knot diagram by overlaying the two dual Tait graphs of the knot diagram. This setting also makes evident applications to Chebyshev or harmonic knots, whose related bipartite graph is the popular grid graph, and to discrete Morse functions. Furthermore we prove structural properties of the bipartite …
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