Narrow Results

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.

The image of a polygonal knot K under a spherical inversion of ℝ³ ∪ ∞ is a simple closed curve made of arcs of circles, perhaps some line segments, having the same knot type as the mirror image of K. But suppose we reconnect the vertices of the inverted polygon with straight lines, making a new polygon . This may be a different knot type. For example, a certain 7-segment figure-eight knot can be transformed to a figure-eight knot, a trefoil, or an unknot, by selecting different inverting spheres. Which knot types can be obtained from a given original polygon K under this process? We show that for large n, most n-segment knot types cannot be reached from one initial n-segment polygon, using a single inversion or even the whole Möbius group.

The number of knot types is bounded by the number of complementary domains of a certain system of round 2-spheres in ℝ³. We show the number of domains is at most polynomial in the number of spheres, and the number of spheres is itself a polynomial function of the number of edges of the original polygon. In the analysis, we obtain an exact formula for the number of complementary domains of any collection of round 2-spheres in ℝ³. On the other hand, the number of knot types that can be represented by n-segment polygons is exponential in n.

Our construction can be interpreted as a particular instance of building polygonal knots in non-Euclidean metrics. In particular, start with a list of n vertices in ℝ³ and connect them with arcs of circles instead of line segments: Which knots can be obtained? Our polygonal inversion construction is equivalent to picking one fixed point p ∈ ℝ³ and replacing each edge of K by an arc of the circle determined by p and the endpoints of the edge.

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.

Kirchhoff energy is a classical functional on the space of arclength-parameterized framed curves whose critical points approximate configurations of springy elastic rods. We introduce a generalized functional on the space of framed curves of arbitrary parameterization, which model rods with axial stretch or cross-sectional inflation. Our main result gives explicit parameterizations for all periodic critical framed curves for this generalized functional. The main technical tool is a correspondence between the moduli space of shape similarity classes of closed framed curves and an infinite-dimensional Grassmann manifold. The critical framed curves have surprisingly simple parameterizations, but they still exhibit interesting topological features. In particular, we show that for each critical energy level there is a one-parameter family of framed curves whose base curves pass through exactly two torus knot types, echoing a similar result of Ivey and Singer for classical Kirchhoff energy. In contrast to the classical theory, the generalized functional has knotted critical points which are not torus knots.

We prove the analyticity of smooth critical points for O’Hara’s knot energies ${{ℰ}}^{\alpha ,p}$, with $p=1$ and $2<\alpha <3$, subject to a fixed length constraint. This implies, together with the already established regularity results for O’Hara’s knot energies, that bounded energy critical points of ${{ℰ}}^{\alpha ,1}$ 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 ${{ℰ}}^{2,1}$.