Tangent-Point Self-Avoidance Energies for Curves

Abstract

We study a two-point self-avoidance energy E q which is defined for all rectifiable curves in R n as the double integral along the curve of 1/r q . Here r stands for the radius of the (smallest) circle that is tangent to the curve at one point and passes through another point on the curve, with obvious natural modifications of this definition in the exceptional, non-generic cases. It turns out that finiteness of E q (γ) for q ≥ 2 guarantees that γ has no self-intersections or triple junctions and therefore must be homeomorphic to the unit circle S 1 or to a closed interval I. For q > 2 the energy E q evaluated on curves in R 3 turns out to be a knot energy separating different knot types by infinite energy barriers and bounding the number of knot types below a given energy value. We also establish an explicit upper bound on the Hausdorff-distance of two curves in R 3 with finite E q -energy that guarantees that these curves are ambient isotopic. This bound depends only on q and the energy values of the curves. Moreover, for all q that are larger than the critical exponent q crit = 2, the arclength parametrization of γ is of class C 1,1−2/q , with Hölder norm of the unit tangent depending only on q, the length of γ, and the local energy. The exponent 1 − 2/q is optimal.Mathematics Subject Classification (2000): 28A75, 49Q10, 53A04, 57M25

A simple scheme for the approximation of self-avoiding inextensible curves

“…1.3] have shown that any curve for which (2) is finite is a C 1,1−2/q manifold. Adapting the arguments in [31] one may relax the condition u ∈ C to u ∈ W 1,1 with positive lower and finite upper bound on |u |.…”

“…The tangent-point energies have been proposed as a family of self-avoiding functionals by Gonzalez and Maddocks [15]; the scale invariant case q = 2 (which we will disregard in this paper) already appears in a paper by Buck and Orloff [8]. They are defined on (smooth) embedded curves u : I → R n and take values in [0, +∞], see Strzelecki and von der Mosel [31] and references therein. Blatt [6] has characterized the energy spaces in terms of Sobolev-Slobodeckiȋ spaces; regularity aspects are discussed in [7].…”

On some knot energies involving Menger curvature

“…Then, there exist constants δ = δ(p) ∈ (0, 1), α = α(p) > 0, β = β(p) > 0 and c(p) < ∞ (all four depending only on p) such that γ has the (d 0 , ϕ)-diamond property for each couple of numbers (d 0 , ϕ) satisfying The proof of this proposition can be easily obtained from our earlier work (see [58,Section 2] for the case of M p , [57, Section 3] for the case of I p , [64,Section 4] for the case of E p ) and Kampschulte's master's thesis [32] for the case of E sym p . The last case of U p can be treated via an application of [58, Remark 7.2 and Theorem 7.3], as the finiteness of U p (γ) for p > 1 and a simple curve γ ∈ C implies, by Hölder inequality,…”

“…x,y,z→ζ R −1 (x, y, z) = κ γ (ζ), so that no regularization is necessary as pointed out by Banavar et al in [3] 4 . Moreover, using the elementary geometric definition of the respective integrands we have gained detailed insight in the regularizing effects of Menger curvature energies in a series of papers [32,57,58,61,64]. In particular, the uniform C 1,α -a-priori estimates for supercritical values of the power p, that is, for p above the respective critical value, for which the corresponding energy is scale-invariant, turn out to be the essential tool, not only for compactness arguments that play a central role in variational applications, but also in the present knot-theoretic context; see Section 2.…”

“…As in Theorem 1.2 of[64] it is actually not necessary to assume equal length of γ 1 and γ 2 .PROOF: Fix η = 1 3 d 0 and pick N > 1/η ≥ N − 1, so that t i := (i − 1)η ∈ [0, 1], i = 1, .. . , N + 1, with the standard convention t N +1 = t 1 yield an equidistant partition of [0, 1].…”