2001/05/16 by Aaron Abrams, Abrams, Aaron, Jason Cantarella +7 · 1 citation
Mathematics · #52A40 #53A04 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT) #math.DG #math.GT #msc:52A40 #msc:53A04
paper · pdf · doi:10.48550/arxiv.math/0105138
15 pages with 3 figures. See also http://www.math.sc.edu/~howard/
arxiv created 2001/05/16 · arxiv updated 2009/11/30
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 G. Luko on average chord lengths of closed curves.