---
abstract: 'In this article we study the regularity of stationary points of the knot energies ${\ensuremath{E^{({\ensuremath{\alpha}})}}}$ introduced by O’Hara in [@oha:en; @oha:fam-en; @oha:en-func] in the range ${\ensuremath{\alpha}}\in(2,3)$. In a first step we prove that ${\ensuremath{E^{({\ensuremath{\alpha}})}}}$ is $C^1$ on the set of all regular embedded curves belonging to ${\ensuremath{H^{\scriptstyle ({\ensuremath{\alpha}}+1)/2,2}}}{\ensuremath{({\ensuremath{\mathbb{R}}}/{\ensuremath{\mathbb{Z}}},{\ensuremath{\mathbb{R}}}^n)}}$ and calculate its derivative.  After that we use the structure of the Euler-Lagrange equation to study the regularity of stationary points of ${\ensuremath{E^{({\ensuremath{\alpha}})}}}$ plus a positive multiple of the length.  We show that stationary points of finite energy are of class $C^\infty$ — so especially all local minimizers of ${\ensuremath{E^{({\ensuremath{\alpha}})}}}$ among curves with fixed length are smooth.' 
author:
- 'Simon Blatt[^1]'
- 'Philipp Reiter[^2] [^3] [^4]'
title: 'Stationary Points of O’Hara’s Knot Energies'
---

[10]{}

A. Abrams, J. Cantarella, J. H. G. Fu, M. Ghomi, and R. Howard.  Circles minimize most knot energies. ,  42(2):381–394, 2003. 

W. Alt, D. Felix, [Ph]{}.  Reiter, and H. von der Mosel.  Energetics and dynamics of global integrals modeling interaction between stiff filaments. ,  59(3):377–414, 2009. 

S. Blatt.  Boundedness and regularizing effects of [O’Hara]{}’s knot energies.  To appear in [*J. Knot Theory Ramifications*]{}, 2011. 

S. Blatt.  The energy spaces of the tangent point energies.  Preprint, 2011. 

S. Blatt.  The gradient flow of [O’Hara]{}’s knot energies.  In preparation. 

S. Blatt.  The gradient flow of the [M]{}öbius energy near local minima. ,  43(3):403 –439, 2012. 

S. Blatt and [Ph]{}.  Reiter.  Does finite knot energy lead to differentiability? ,  17(10):1281 – 1310, 2008. 

F. M. Christ and M. I. Weinstein.  Dispersion of small amplitude solutions of the generalized [K]{}orteweg-de [V]{}ries equation. ,  100(1):87–109, 1991. 

R. R. Coifman and Y. Meyer. ,  volume 57 of [*Astérisque*]{}.  Société Mathématique de France, Paris, 1978. 

M. H. Freedman, Z.-X. He, and Z. Wang.  Möbius energy of knots and unknots. ,  139(1):1–50, 1994. 

S. Fukuhara.  Energy of a knot.  In [*A fête of topology*]{}, pages 443–451.  Academic Press, Boston, MA, 1988. 

L. Grafakos. ,  volume 250 of [*Graduate Texts in Mathematics*]{}.  Springer, New York, second edition, 2009. 

Z.-X. He.  The [E]{}uler-[L]{}agrange equation and heat flow for the [M]{}öbius energy. ,  53(4):399–431, 2000. 

J. O’Hara.  Energy of a knot. ,  30(2):241–247, 1991. 

J. O’Hara.  Family of energy functionals of knots. ,  48(2):147–161, 1992. 

J. O’Hara.  Energy functionals of knots. [ II]{}. ,  56(1):45–61, 1994. 

J. O’Hara. ,  volume 33 of [ *Series on Knots and Everything*]{}.  World Scientific Publishing Co. Inc., River Edge, NJ, 2003. 

.  Reiter.  Regularity theory for the [M]{}öbius energy. ,  9(5):1463–1471, 2010. 

.  Reiter.  Repulsive knot energies and pseudodifferential calculus for [O’Hara]{}’s knot energy family ${E}^{(\alpha)}$. To appear in *Mathematische Nachrichten*, [DOI: 10.1002/mana.201000090](http://dx.doi.org/10.1002/mana.201000090) (2012). 

T. Runst and W. Sickel. ,  volume 3 of [*de Gruyter Series in Nonlinear Analysis and Applications*]{}.  Walter de Gruyter & Co., Berlin, 1996. 

P. Strzelecki, M. Szuma[ń]{}ska, and H. von der Mosel.  A geometric curvature double integral of [M]{}enger type for space curves. ,  34(1):195–214, 2009. 

P. Strzelecki, M. Szuma[ń]{}ska, and H. von der Mosel.  Regularizing and self-avoidance effects of integral [M]{}enger curvature. ,  IX(1):145–187, 2010. 

P. Strzelecki and H. von der Mosel.  Integral [M]{}enger curvature for surfaces. ,  226(3):2233–2304, 2011. 

P. Strzelecki and H. von der Mosel.  Tangent-point self-avoidance energies for curves.  To appear in *Journal of Knot Theory and its Ramifications*, [DOI: 10.1142/S0218216511009960](http://dx.doi.org/10.1142/S0218216511009960) (2011). 

M. E. Taylor. ,  volume 115–117 of [*Applied Mathematical Sciences*]{}.  Springer-Verlag, New York, 1996–1997. 

[^1]: Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL, United Kingdom, [S.Blatt@warwick.ac.uk](S.Blatt@warwick.ac.uk)

[^2]: Abteilung für Angewandte Mathematik, Universität Freiburg, Hermann-Herder-Straße 10, 79104 Freiburg i. Br.,  Germany, [reiter@mathematik.uni-freiburg.de](reiter@mathematik.uni-freiburg.de)

[^3]: MSC (2000) 42A45, 53A04, 57M25

[^4]: The final publication is available at [www.springerlink.com](www.springerlink.com).