Circles minimize most knot energies

By Aaron Abrams, Jason H. Cantarella, Joseph H. G. Fu, M. Ghomi, R. Howard. 2001

73 citations recorded by the source corpus.

Abstract We define a new class of knot energies (known as renormalization energies ) and prove that a broad class of these energies are uniquely minimized by the round circle. Most of O'Hara's knot energies belong to this class. This proves two conjectures of O'Hara and of Freedman, He, and Wang. We also find energies not minimized by a round circle. The proof is based on a theorem of Lukő on average chord lengths of closed curves.

Subfields: Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/5b13075e989e8f51136a479cde820ab996845f56

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

Citations in the corpus

On rectifiable curves with Lp-bounds on global curvature: self-avoidance, regularity, and minimizing knots (2007, background): “We recall the definition of knot or isotopy classes in R3 : Two continuous closed curves γ1, γ2 ⊂ R3 are isotopic, denoted as γ1 γ2, if there are open neighbourhoods N1 of γ1,N2 of γ2, and a continuous mapping : N1 × [0, 1] → R3 such that (N1, τ) is homeomorphic to N1 for all τ ∈ [0, 1], (x, 0) = x ”