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Presentation files (TeX prosper, beamer)
  1. A survey of ``Energy of knots and conformal geometry" , pdf file of 40 pages color OHP sheets, (about 740KB)(Dec. 2003)
  2. Infinitesimal cross ratio and conformally invariant energies for knots, a survey of a part of the second half of the book. (pdf file of transparancy sheets with color)(320KB)(March 2003)
  3. Conformal geometry of curves(2 pages)(May 2005)
  4. Conformal geometry of curves and surfaces (August 2008); for presentation and for printing.
  5. COE Conference Groups, ``Homotopy and Configuration Spaces''(07/2005)(2.9MB)
  6. Conformal geometry of knots(02/2006)(3.6MB)
  7. Mobius geometry of the set of spheres(03/2006)(4.6MB)
  8. The structure of the set of spheres and its applications(03/2007)(4.2MB)
  9. The spaces of subspheres and their applications (7MB), The spaces of subspheres and their applications (a little bit longer 7.2MB), The spaces of subspheres and their applications (for printing) (2.5MB), The spaces of subspheres and their applications (2008) (8.4MB)
  10. Configuration space of equilateral and equiangular n-gons (n<=6) (08/2008) (3MB) (Knots and soft-matter physics, 26-29 Aug. 2008, Kyoto Univ.)
  11. Renormalization of r*-potentials and generalized centers (06/2011), Differential Geometry and Parametrization of 3D Knots, Centro di Ricerca Matematica (CRM), Pisa (Italy)
  12. Mobius invariant energies and average linking with circle (07/2011), ESF-EMS-CRM-Pi International Conference on Knots and Links: From Form to Function, Centro di Ricerca Matematica (CRM), Scuola Normale Superiore, Pisa (Italy)

Pdf files of OHP transparancies and survey articles

* Energy fo knots(Survey article) pdf file (772KB) (22 Jan 2000)
* A short abstract of a small part of the book in 2 pages, pdf file (Sept. 2002)

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  • (with R. Langevin) Conformally invariant energies of knots , J. Institut Math. Jussieu 4 (2005), 219-280, ps and pdf files, abstract
  • The configuration space of planar spidery linkages (revised June 2006) ArXive, pdf file with figures with higher resolutions, Topology Appl 154 (2007), 502-526.
  • A Note on Y-energies of Knots, OCAMI Studies Vol 1. Knot Theory for Scientific Objects, proceedings of the International Workshop on Knot Theory for Scientific Objects.
  • Energy of knots and the infinitesimal cross ratio, Proceedings of the Conference "Groups, Homotopy and Configuration Spaces", Geometry and Topology Monographs (2007).
  • Area of a product torus of a link (2007)
  • Conformal dual of a quadruplet of points Far East Journal of Mathematical Education 2 (2008) 1-11
  • (with R. Langevin) Conformal invariance of the writhe of a knot to appear in J. Knot Theory Ramifications
  • (with R. Langevin) Conformal arc-length as 1/2 dimensional length of the set of osculating circles to appear in Comm. Math. Helv.
  • Renormalization of $r^{\bullet}$-potentials and generalization of dual volumes and centers
  • (with Gil Solanes) Mobius invariant energies and average linking with circles
  • The incenter of a triangle as a cone isoperimetric center
  • (with R. Langevin and S. Sakata) Space of subspheres and conformal invariants of curves
  • Ideal, best packing, and energy minimizing double helices
  • The configuration space of equilateral and equiangular hexagons


  • Programm of energy of knots by K. Ahara (Meiji Univ.) (text file)