Dynamics of Embedded Curves by Doubly-Nonlocal Reaction-Diffusion Systems
We study a class of nonlocal, energy-driven dynamical models that govern the motion of closed, embedded curves from both an energetic and dynamical perspective. Our energetic results provide a variety of ways to understand physically motivated energetic models in terms of more classical, combinatorial measures of complexity for embedded curves. This line of investigation culminates in a family of complexity bounds that relate a rather broad class of models to a generalized, or weighted, variant of the crossing number. Our dynamic results include global well-posedness of the associated partial differential equations, regularity of equilibria for these flows as well as a more detailed investigation of dynamics near such equilibria. Finally, we explore a few global dynamical properties of these models numerically.
In a variety of physical and mathematical contexts, a curvature-regularized nonlocal interaction energy
governs the motion of a closed, embedded curve. For example, worm-like chain (WLC) models for DNA dynamics incorporate the bending energy (i.e. the squared curvature ) and an energetic barrier to self-intersection (i.e. a repulsive kernel ) that prevents topological changes. Dynamical models of protein folding  and vortex filaments  also take a similar form. At the purely mathematical end of the spectrum, a relatively recent trend in geometric knot theory has witnessed a growth in the energetic study of (1) and its variants ; ; ; ; . A significant portion of this general line of work focuses on extremal properties of these various knot energies such as the existence, complexity and regularity of extremal embeddings within a given class of curves. In the context of (1), the bi-variate kernel encodes the interaction between pairs of points on a curve according to the principles of physical potential theory — an attraction or repulsion between points generically occurs in regions where increases or decreases in its first argument, respectively. A typical kernel will exhibit a singular, short-range repulsion and possibly an additional regular, far-field attraction, yet even this reasonable level of generality in the choice of kernel leads to a class of models that exhibit a striking variety of complex and intricate dynamics.
We study the energetics and dynamics of (1) from both an analytical and a numerical perspective. At an energetic level, we establish a set of inequalities that relate the nonlocal energy (1) to classical, combinatorial measures of complexity for embedded curves. These results prove similar, in spirit, to a variety of work on “knot energies” that emanates from the knot theory community ; ; ; ; ; . Our arguments reveal the sense in which some combinatorial measures of complexity, such as the average crossing number, have fractional Sobolev spaces, lurking just underneath the surface. This insight allows us to establish complexity bounds for a rather broad class of kernels and it also provides the impetus to introduce generalizations of such complexity measures. In this sense, our work complements that strand of geometric knot theory that emphasizes the importance of taking a more analytic approach to (1) and its variants ; ; . For instance, the relative importance of harmonic analysis vis-à-vis Möbius invariance has been observed when studying regularity of extremal embeddings . At a dynamic level, we study a corresponding “constrained gradient flow” of the energy (1) that restricts the dynamics to lie in the class of unit speed parameterizations. This is a slightly non-standard approach since we do not simply functionally differentiate (1) with respect to , but it leads to an analytically and numerically well-behaved dynamical model. In particular, enforcing a uniform density along at each time yields semi-linear rather than quasi-linear dynamics and also gives uniform numerical samples along at each numerical time-step without having to resort to re-parameterizations. At a physical level, we may view our choice of dynamics as a hard limit of WLC models that include energetic barriers to bond stretching, in the sense that an infinite barrier to bond stretching results from constraining the flow to lie in the class of unit speed curves. This approach to dynamics leads to a class of nonlocal reaction-diffusion systems driven by doubly-nonlocal forcings. The pairwise interaction energy leads to a forcing whose derivative along satisfies
and so recovering from requires two consecutive nonlocal operations. We refer to the resulting class of dynamics as a doubly-nonlocal reaction-diffusion system to emphasize this structure. Overall, this approach to dynamics allows us to obtain a global well-posedness result under quite general hypotheses on to demonstrate regularity of critical embeddings, to perform a detailed study of equilibria and to devise reliable numerical procedures.
Most of our arguments rely, at least in part, on some basic harmonic analysis and a few notions from knot theory; we begin by reviewing this material in the next section. We then proceed to study the energy (1) from a complexity point-of-view in the third section. As an example in this direction, a corollary of our analysis shows that the distortion and bending energy bound the average crossing number
of an embedding. An analogous result holds for (1) provided satisfies a “homogeneity” assumption. This is perhaps surprising in light of the following observation: There exists an infinite sequence of ambient isotopy classes of knots with uniformly bounded distortion as well as an infinite sequence of ambient isotopy classes of knots with uniformly bounded bending energy. These sequences therefore exhibit “conflicting” behavior, since (2) rules out the existence of such an infinite sequence when both distortion and bending energy remain uniformly bounded. We also show that an energy of the form (1) induces, in a quantified sense, a stronger invariant than the average crossing number. These results are motivated by analogous results for the Möbius energy  (i.e. (1) with and without curvature) and the ropelength ; ; ; ; ; however, our results are not entirely comparable with these earlier works and we generally prove them using different means. We then proceed to address the dynamics of (1) in the fourth section. We initiate this process by proving global well-posedness and regularity under certain “homogeneity” and “degeneracy” assumptions on the kernel . These assumptions cover many of the kernels that have generated prior interest in the literature ; ; ; , and the main contribution here lies in finding a relatively general and broadly applicable hypothesis under which such a global result holds. This result complements prior work that proves global existence for (1) under the Möbius kernel , although we assign dynamics in a slightly different fashion. We conclude the fourth section with an analysis of dynamics near equilibrium. For instance, it is known that unit circles globally minimize (1) under proper monotonicity and convexity assumptions , and in addition, that desirable regularity properties (e.g. smoothness) hold for critical points of the O’Hara and Möbius energies ; . We supplement this by adapting techniques for nonlocal dynamical systems to show that circles are also properly “isolated” modulo to rotation and scaling invariance. This yields a local asymptotic stability result for our dynamics, which should be compared to  and the recent contribution  for the special case of pure Möbius flow. We use a similar technique to show that the natural “global” version of this result fails for a pure bending energy flow: There exist unknotted initial configurations that remain unknotted for all time under a bending energy flow, yet the dynamics do not asymptotically converge to the globally minimal, unknotted circle. This dynamic result complements the energetic result, proved in , that the double-covered circle the unique bending energy minimizer for the trefoil knot type. Finally, we explore global dynamical properties numerically in the final section.
To place these efforts in context, note that a large body of work has studied the energetics and dynamics of various knot energies. Much of the work in the field has focused on a few families of knot energies such as those suggested in , including tangent-point energies ; , Menger curvature energies  and O’Hara’s energies ; ; ;  (of which the Möbius energy  is a special case). The most closely related examples include work on the energetics and minimizers of integral Menger curvature  , as well as work on the energetics and criticality theory for the Möbius energy ; ; ;  and the extended O’Hara family ;  . In contrast to these energies, the physical model (1) crucially depends on the bending energy which, as previously noted , obviously improves the analytic properties of the total energy. When taken together our results show that, in addition to its physical significance, the admission of the bending energy and a (quite weak) barrier to self-intersection in (1) allows for a wide class of models that exhibit all of the analytic features that have previously motivated knot energies.
2 Preliminary Material
We begin our study by introducing the notation we shall use, then providing a few key definitions and finally recording a series of preliminary lemmas on which we will rely throughout the remainder of the paper. Lightface roman letters such as and will denote real numbers; their boldface counterparts and will denote three-dimensional vectors. We reserve for the canonical standard basis. Given we use for the standard Euclidean inner-product, for the cross product and for the Euclidean norm. The notation denotes the scalar triple product. We use to denote the standard one-dimensional torus, which we view as the interval with endpoints identified. For we define
and then extend to all of via -periodicity. Thus for any pair of points the quantity simply gives the geodesic distance between them. Given a square-integrable function we shall always use the definition and notation
to denote the -norm, with the dimensionality omitted but always clear from context. We shall use
to denote the forward and inverse Fourier transforms, so that
provide the norm as above and an equivalent definition of the Sobolev norm. For any sufficiently regular mapping we use or to denote ordinary differentiation in both the strong and weak sense. We say a rectifiable has unit speed if on and we then set or simply as the pointwise curvature of such an embedding. Similarly, we say a rectifiable has constant speed if , where
denotes the length of the embedded curve. Conversely, given any tangent field with and we may define
and thereby recover a -periodic, constant speed curve that has and center of mass at the origin. We refer to as the knot induced by such a vector-field. Given mean zero function we analogously use
to denote its mean zero primitive.
In addition to these notations, we also need to recall a few elementary facts from harmonic analysis. First, for a given we use the notation
to denote its maximal function, see, for example, (, p.216). We then recall the standard fact that the map defines a bounded operator on , so that
for some absolute constant. If has unit speed then furnishes its pointwise curvature, in which case we define and refer to as the curvature maximal function. The following elementary lemma will also prove useful —
Assume that and that there exist finite constants so that the decay estimates
hold. Then the product obeys the decay estimate
for a universal constant. If obey the decay estimates
where are finite constants and are arbitrary exponents, then the product obeys the decay estimate
with a positive constant depending only upon the exponents.
See  lemma 6.1 and lemma 6.2, for instance, for an indication of the proof.
Recall that a unit speed embedding is bi-Lipschitz if there exists a constant so that the inequality
holds for any possible pair of points. Thus exactly when Gromov’s distortion
is finite, with furnishing the smallest possible bi-Lipschitz constant. Moreover, the lower bound holds for any closed, rectifiable curve with unit speed, see pp.6-9 . We shall refer to any pair of points for which
as a distortion realizing pair. A simple invocation of Taylor’s theorem shows that
for any with unit speed, and thus any such actually admits a distortion realizing pair. Given a bi-Lipschitz embedding with unit speed, we define
as the maximal distance on between any distortion realizing pair. By applying the first derivative test to the expression we obtain
and so by applying Taylor’s theorem and the triangle inequality to bound the right hand side of this expression from below we obtain a lower bound
for the distance on between any such pair. Finally, if a bi-Lipschitz embedding has unit speed and if we define as
we may appeal to Taylor’s theorem one final time to conclude that the inequalities
hold. Despite their elementary proofs, these inequalities prove quite useful.
Finally, for given a bi-Lipschitz embedding we shall use to denote a generic ambient isotopy class. Similarly, we shall employ while the notation if we wish to emphasize the class induced by some underlying curve. The following lemma from  and the embedding shows that ambient isotopy classes are well-behaved with respect to convergence in both the and topologies —
Let denote a positive velocity, simple closed curve. Then there exists such that all satisfying are ambient isotopic. In particular, if is sufficiently small.
3 Basic Energetics
This section provides a brief analysis of nonlocal energies that, when defined over unit speed embeddings, take the form
for some bi-variate kernel. We shall also consider the normalized or scale-invariant bending energy
The motivation for these choices arises, at least in part, from the fact then defines a differentiable approximation of the distortion. More specifically, prior work ;  demonstrates that necessarily implies that has finite distortion. Moreover, the O’Hara family converges as and to the log-distortion of after a suitable normalization. This observation suggests the somewhat more obvious family
since then corresponds to the classical -norm approximation of the -norm. Further motivations for using (9) include the fact that the Möbius energy and a large class of the O’Hara energies attain their global minimum at the standard embedding of the unit circle, as well as the fact that the Möbius energy exhibits a relationship with classical combinatorial quantities such as the crossing number.
We now show that a very large class of energies based on have these three motivating properties. In a certain sense these properties are best understood from a pure analytical point-of-view, rather than from an appeal to geometric or topological considerations (e.g. Möbius invariance). This point of emphasis echoes an observation made in earlier work on the Möbius energy — the smoothness of its critical points follows without explicitly appealing to Möbius invariance itself . We begin by showing that, provided exhibits a sufficient degree of singularity, finiteness of the integral (8) necessarily implies that an curve is bi-Lipschitz. We shall also obtain a concrete bound for the distortion in terms of and the semi-norm in the process, although the bound itself is far from optimal for general kernels.
Assume that satisfies the homogeneity property
for some exponent and some function . Assume that the lower bound holds on as well. If has unit speed then
for finite constants. Moreover, if then
and so is bi-Lipschitz if is finite.
See appendix, lemma A.1 ∎
The homogeneity hypothesis provides a means to unify and simplify our analysis, yet it proves general enough to cover all families introduced so far. For instance, we have and for the Möbius energy, the relations and for the O’Hara family and and for the -distortion. The lemma therefore applies for all three families, provided in the second case and in the final case. It is also clear that the conclusion holds for kernels of the form with an homogeneous kernel and bounded from below, although we have no impetus to pursue this level of generality since the requisite modifications to the argument and its conclusion are straightforward. Finally, we cannot remove the dependence on or dispense with the hypothesis that and still have the conclusion of the lemma hold at this level of generality. Indeed, for the -distortion family it is easy to construct a smooth sequence for which remains uniformly bounded but and diverge.
The previous lemma illustrates the fact that a wide class of energies
approximate the distortion. Under similar hypotheses on the kernel both the energy and the bending energy attain their global minima at the standard embedding of the unit circle ; . We shall quickly review the arguments underlying these known facts, as this discussion will allow us to emphasize an analytical point — both arguments appeal to the same classical technique in the calculus of variations, i.e. the use of Poincarè-type inequalities with optimal constants. For instance, the statement follows directly from the Poincarè-type inequality
whenever decreases in its first argument over the non-negative reals. By Jensen’s inequality this in turn shows that
whenever is also convex. Demonstrating global optimality of for the bending energy proceeds in a similar fashion. In this case the classical Poincarè inequality with optimal constant
provides the starting point. Letting denote the unit speed embedding of a simple appeal to parametrization and scale invariance of shows
Suppose that for each the function is non-decreasing and convex on . Then for any the energy
attains its minimum over unit speed curves at the unit circle. Moreover, if either or is strictly decreasing then unit circles are the unique global minimizers.
In particular, the Möbius energy as well as the O’Hara family and the -Distortion family all have as their global minimizer. Once again, the arguments above leading to this theorem are both standard and known ; ; ; we recall them simply to emphasize the connection between them as well as the broader connection to classical PDE and variational arguments. Thus there are analytical properties, such as convexity and Sobolev inequalities, rather than geometric properties, such as Möbius invariance, lying at the heart of the matter.
Finally, we turn our attention toward relating the energies and to more classical measures of complexity. Once again, analytical considerations and Sobolev spaces shall come to the fore. A corollary of this analysis will also allow us to illustrate that an energy of the form induces, in a certain sense, a much stronger invariant than the crossing number. Following ; ; , we shall begin by considering the mapping defined by
with a Lipschitz embedding with finite distortion. For any such Lipschitz curve the function is Lipschitz in both variables whenever and has finite distortion. For such functions, the co-area formula for Lipschitz maps  therefore implies that
provided is a positive, Lebesgue measurable function. For a given let us define the crossing set as
where represents the corresponding planar curve induced by orthogonal projection. Thus precisely when distinct points of self-intersect. As the equality
holds by a trivial calculation, we may therefore conclude
When the integrand
in (13) simply gives the cardinality of and so (13) reduces to a constant multiple of the average crossing number. We may utilize the non-negative function to apply a positive weight applied to each point of self-intersection in and in this way arrive at a weighted generalization or “weighted” crossing number. By analogy with electrostatics, we shall consider the simple power-law family
of weighting functions, where denotes the arc-length between points. We may then view (13) as inducing a “repulsion” between points of self-intersection, in the sense that small values occur when weighted crossings are, on average, equally spaced along as measured by relative length. We primarily intend this family as an analytical device or gauge for measuring and drawing analytical comparsions — a bound on (13) for represents a stronger conclusion, in general, than a bound on the crossing number or average crossing number. Moreover, for smooth curves Taylor’s theorem immediately yields
represents the strongest possible conclusion, and therefore the upper limit of our analytical scale.
We shall need with the following lemma in order to perform our analytical comparison.
Assume that and that has finite distortion. Assume further that has constant speed. Then
and if then the constant
See appendix, lemma A.2 ∎
This lemma allows us to relate combinatorial measures of complexity for a curve to more standard measures of regularity of the embedding. By taking and evaluating the constant explicitly we may observe the most straightforward consequence, i.e. the inequality
for the average crossing number. To help place this inequality in a more familiar context, given an exponent and a rectifiable let us define the scale-invariant total -curvature of as
The case yields the (square root of) the bending energy, while the case reduces to Milnor’s notion of total curvature . For constant speed curves and the inequality
holds, for some positive constant, due to the Hausdorff-Young inequality. If we may therefore conclude the following total -curvature bounds
for the average crossing number. From this observation we conclude that essentially any assumption whatsoever that is stronger than an assumption of finite total curvature yields a bound for the average crossing number. Indeed, instead of assuming for some we could also invoke the slightly weaker assumption that lies in the real Hardy space and still deduce an average crossing number bound. As the constants do not remain bounded, however, and so the natural “limiting” conclusion of (18) need not hold in general. The Hardy space usually serves as the substitute for in such a circumstance, but as (17) reveals, the fractional Sobolev space is actually the natural choice: The inequality (17) recovers the “proper” limiting inequality as , for while neither embedding
holds, the finiteness of implies the finiteness of both. As we recover the norm rather than finite total curvature in (17), and it is therefore the natural measure of regularity to use. It prefers oscillatory components in the curvature measure rather than the piecewise-constant curvature measures that characterize embeddings with finite total curvature, and so embeddings with oscillatory curvature rather than piecewise-linear embeddings must necessarily have finite average crossing number.
Another corollary of the bound (17) shows that finite bending energy and finite distortion imply a finite average crossing number. In fact we obtain a stronger conclusion from this analysis, in that an assumption of finite bending energy yields a significantly stronger conclusion than a simple crossing number bound. Specifically, for any the weighted crossing number
remains bounded whenever has finite distortion and finite bending energy. To appreciate the significance of this bound, recall that Gromov has provided an infinite sequence of ambient isotopy classes that have uniformly bounded distortion (, p.308). In addition, every -torus knot has a smooth unit-length embedding in such that for every , and so there also exists an infinite sequence of ambient isotopy classes with uniformly bounded bending energy. The crossing number bounds the weighted crossing number from below and defines a finite-to-one invariant (i.e. for every fixed integer there are at most finitely many knots with crossing number equal to ), so (19) cannot hold if we neglect either the bending energy or the distortion. However, by combining them we obtain not only a crossing number bound, but in fact a stronger weighted crossing number bound. Moreover, (19) shows that no infinite sequence of embeddings of distinct ambient isotopy classes of knots has both uniformly bounded distortion and uniformly bounded bending energy. A similar bound applies for whenever the kernel satisfies lemma 3.1, and so the energy is stronger than the weighted crossing number in an analogous sense.
Finally, it is worth briefly mentioning the consequences of (19) at the level of invariants. We may use or the weighted crossing number to define invariants via minimization in the standard way, i.e. by defining
for an arbitrary ambient isotopy class. That (under appropriate hypotheses on the kernel) and actually attain their minima within an ambient isotopy class follows from lemma 2.2 and a standard argument based on the direct method. For these invariants a bound of the form
holds for arbitrary. By appealing to results relating the curvature and distortion to ropelength ; ; ;  and ropelength to crossing number  we may also conclude the corresponding upper bound
and so these invariants are polynomially equivalent.
4 Dynamics of Embedded Curves
With a basic understanding of the energies established, we now turn our attention toward the dynamics they induce via the constrained gradient flow
of such an energy. Given a bi-variate kernel we shall use
to denote the self-repulsive nonlocal forcing. For any tangent field defining the Lagrange multipliers and as
completes the description of the flow. The presence of these multipliers guarantees that and for all provided these properties hold initially. The corresponding induced knot