Discrete thickness

Abstract

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 → ∞, regarding the topology induced by the Sobolev norm || · || W 1,∞ (S 1 ,R d ) . This result directly implies the convergence of almost minimizers of the discrete energies in a fixed knot class to minimizers of the smooth energy. Moreover, we show that the unique absolute minimizer of inverse discrete thickness is the regular n-gon.

New Directions in Geometric and Applied Knot Theory

“…This has various consequences in geometric knot theory, for instance, it completes Scholtes' recent investigations on a discrete version of the Möbius energy for polygons with n edges, that can now be shown to Γ-converge to the Möbius energy (2.1.1) as n → ∞; see [105,Theorem 1.1] and [14,Theorem 3.8]. We do not address the very interesting questions regarding suitable discretisations and merely refer to the work of Rawdon et al [92,93,95,78,94,96,97,106] on discretised versions of ropelength, and to [108,105,107] for discretisations of a few other geometric curvature energies.…”

“…in the class of two-dimensional embedded surfaces with prescribed genus or under alternative constraints; see also [10,100,111,69,58,99,79]. But minimising the Willmore energy or related functionals such as the Helfrich functional [51,81] on given isotopy classes has to the best of our knowledge not been investigated yet -with the exception of recent work of P.…”

Tangent-point energies and ropelength as Gamma-limit of discrete tangent-point energies on biarc curves

“…This result was strengthened later by Blatt [6]. In [26,27] he proved the Γ-convergence of polygonal versions of ropelength and of integral Menger curvature to ropelength and to continuous integral Menger curvature, respectively. It remains open at this point if stronger types of variational convergence such as Hausdorff convergence of sets of almost minimizers can be shown for the non-local knot energies treated here, as was, e.g., established in [28] for the classic bending energy under clamped boundary conditions.…”

Discrete knot energies

“…More generally, the result remains true if instead of equilateral polygons one takes the space of polygons with a uniform bound on longest to shortest segment length. These results were then used to prove the convergence of ideal polygonal to smooth ideal knots, a result that could later be improved from C 0 convergence to C 0,1 convergence Corollary 16 (Ideal polygonal knots converge to smooth ideal knots, [Raw03,Sch14b]). Let K be a tame knot class and p n ∈ P n (K) bounded in L ∞ with |inf Pn(K) ∆ −1 n − ∆ n [p n ] −1 | → 0.…”

“…Similar questions for more general energies were considered (see [DD00,Raw03]). If the knot class is not fixed, the unique absolute minimizers of ∆ −1 n is the regular n-gon: Proposition 18 (Regular n-gon is unique minimizer of ∆ −1 n , [Sch14b]). The unique minimizer of ∆ −1 n in P n is the regular n-gon.…”