Discrete Möbius energy

Abstract

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.Mathematics Subject Classification (2010): 49J45; 57M25, 49Q10, 53A04

Discrete knot energies

“…Proposition 8 (Order of convergence for Möbius energy, [Sch14a]). Let γ ∈ C 1,1 (S L , R d ) be parametrised by arc length and c, c > 0.…”

“…Corollary 9 (Convergence of Möbius energies of inscribed polygons, [Sch14a]). Let γ ∈ C with E(γ) < ∞ and p n as in Proposition 8.…”

“…Lemma 10 (Regular n-gon is unique minimizer of discrete Möbius energy, [Sch14a]). The unique minimizer of E n in P n is the regular n-gon.…”

“…Corollary 11 (Convergence of discrete minimizers to the round circle, [Sch14a]). Let p n ∈ P n bounded in L ∞ with E n (p n ) = inf Pn E n .…”

Discrete thickness

“…Proof. In [20,Proposition 10] we showed that if n is large enough we can find an equilateral inscribed closed polygonp n of lengthL n ≤ 1 with n vertices that lies in the same knot class as γ. By rescaling it to unit length via p n (t) = LL −1 npn (L n L −1 t), L = 1, we could show in addition that p n → γ in W 1,2 (S 1 , R d ), as n → ∞.…”