# Research Report

Outline of my research
My motivational problem which was originally proposed by Fukuhara and Sakuma is:

1. Define a functional, "energy of knots", on the space of the knots.
2. Define the canonical shape of any given knot type as the embedding that attains the minimum value of that functional within its ambient isotopy class. (We call it an energy minimizer.)

For this purpose we consider a functional on the space of knots that blows up if a knot degenerates to a 'singular knot' with self-intersections. Such a functional is called a knot energy functional.
The electrostatic energy of a charged knot would be the most naive candidate for that. Although, it blows up for any knot. I could get rid of this difficulty of explosion by so called regularization to have a finite valued functional. But then, this functional does not blow up for a 'singular knot' with double points. By changing the power index of the integrand, which corresponds to a non-physical assumption that Coulomb's repelling force between a pair of point charges is inversely proportional to more than the cube of the distance between them, I defined a family of energy functionals.
My research concerns the problem to determine whether there exists an energy minimizer for each knot type with respect to a given knot energy functional. The answer to this problem depends on the following conditions: the power index of the integrand of the energy functional, the metric of the ambient space, i.e. whether it is Euclidean, spherical, or hyperbolic, and the knot type, i.e. whether it is prime or not.

Langevin defined a new functional on the space of knots from integral geometric viewpoint.
Let S denote the set of 2-spheres in S^3. Then S can be realized as 4 dimensional hyperbolic hypersurface with one sheet in the 5 dimensional Minkowski space. A 4-tuple of points on a given knot K in S^3 defines a (generically unique) 2-sphere that passes through the 4 points. Thus a knot K defines a map from a subset in T^4 to the set of 2-spheres in S^3, S. We call a 2-sphere in the image of this map a non-trivial sphere of the knot K. Langevin's functional, the volume of non-trivial spheres, is defined as the volume with multiplicity of the image of this map. He showed that his functional is a knot energy functional. We gave a formula to express this functional in terms of the infinitesimal cross-ratio.

These results, together with related topics, can be found in my book [11].

Publications
Research papers
[1] Energy of a knot, Topology 30 (1991), 241-247
[2] Family of energy functionals of knots, Topology Appl. 48 (1992), 147-161
[3] Energy functionals of knots, in "Topology -Hawaii", K.H.Dovermann ed., World Scientific, Singapore, (1992), 201-214
[4] Energy funcitonals of knots II, Topology Appl. 56 (1994), 45-61
[5] Energy of knots in a 3-manifold; The spherical and the hyperbolic cases, Proce edings of Knots '96, S. Suzuki ed., World Scientific, 1997, 449-464
[6] Jun Imai, Energy of knots, in Japanese, survey article, Sugaku, 1997, Vol 49-4. 365 - 378
[7] Energy of knots, Sugaku Expositions 13 ,2000, 73-90.
[8] Energy of knots, "Ideal Knots", A. Stasiak, V. Katrich, L. H. Kauffman eds., World Scientific, 1998, 288-314
[9] Asymptotic formulae of energies of polygonal knots, Proceedings of the Conference on Low Dimensional Topology, H. Nencka and R. Vasconcelos eds., Contemporary Mathematics (CONM) book series, Amer. Math. Soc., 1999, 235-249
[10] (with Remi Langevin) Conformally geometric viewpoints for knots and links I, in Physical Knots: Knotting, Linking, and Folding Geometric Objects in R^3, AMS Special Session on Physical Knotting and Unknottin, Las Vegas, Nevada, April 21-22, 2001, J. A. Calvo, K. Millett, and E. Rawdon eds., Contemp. Math. 304, Amer. Math. Soc., Providence, RI, (2002), 187-194.
[12] (with R. Langevin) Conformally invariant energies of knots, J. Institut Math. Jussieu 4 (2005), 219-280. Available at ArXive, abstract.
[13] The configuration space of planar spidery linkages, Topology Appl. 154 (2007), 502-526. Available at ArXive
[14] The configuration space of a spider, Proceedings of the int ernational conference, "Intelligence of Low Dimensional Topology 2006" published by World Scientific Publishing Co. in the Knots and Everything Book Series Vol. 40 (2007) 245-252.
[15] A Note on Y-energies of Knots, OCAMI Studies Vol 1 (1). Knot Th eory for Scientific Objects, proceedings of the International Workshop on Knot Theory for Scientific Objects (2007) 85-95. Available at ArXive.
[16] Energy of knots and the infinitesimal cross ratio, Proceedings of the Conference "Groups, Homotopy and Configuration Spaces", Geometry and Topology Monographs 13(2008) 421-445.
[17] Conformal dual of a quadruplet of points, Far East Journal of Mathematical Education 2 (2008) 1-11. arXiv:0709.0454
[18] (with Remi Langevin) Conformal arc-length as 1/2-dimensional length of the set of osculating circles, Comm. Math. Helv. 85 (2010) 273-312 arXiv:0803.1060
[19] (with Remi Langevin) Conformal invariance of the writhe of a knot, J. Knot Theory Ramifications 19 (2010) 1115-1123, arXiv:0803.1876
[20] (with Udo Hertrich-Jeromin and Alastair King) On the Moebius geometry of Euclidean triangles, Elemente der Mathematik 68 (2013), 96-114.
[21] Ideal, best packing, and energy minimizing double helices, Progress of Theoretical Physics Supplement 191 (2011) 215-224, arXiv:1104.0489v1 [physics.comp-ph]
[22] Renormalization of potentials and generalized centers, Adv. Appl. Math. 48 (2012), 365-392, arXiv:1008.2731 [math.DG]
[22'] Corrigendum to Renormalization of potentials and generalized centers'' [Adv. in Appl. Math. 48 (2) (2012) 365-392], Adv. in Appl. Math. 49 (2012) 397-398 (replacement of a graph of a potential)
[23] Isoperimetric characterization of the incenter of a triangle, Elem. Math. 68 (2013), 78-82.
[24] The configuration space of equilateral and equiangular hexagons, Osaka J. Math. 50, (2013), 477-489, arXiv:1105.5046 Gif movies of equilateral and equiangular hexagons
[25] (with Remi Langevin and Shigehiro Sakata) Application of spaces of subspheres to conformal invariants of curves and canal surfaces, Ann. Polon. Math. 108 (2013), 109-131
[26] Measure of a 2-component link, Tohoku Math. J. Tohoku Math. J. 65 (2013), 427-440arXiv:0709.2215
[27] Minimal unfolded regions of a convex hull and parallel bodies, to appear in Hokkaido Math. J. arXiv:1205.0662
[28] (with Gil Solanes) Mobius invariant energies and average linking with circles, Tohoku Math. J. 67 (2015), 51-82
[29] (with H. Funaba) Mobius invariant energy of tori of revolution, to appear in Proceedings of "Knotted, Linked and Tangled Flux in Quantum and Classical Systems" of the Newton Institute, Tom Kephart and Keith Moffatt eds.

* [15], [17], and [21] are not on the list of MathSciNet.

Books
[11] J. O'Hara, Energy of knots and conformal geometry. Series on Knots and Everything Vol. 33, World Scientific, Singapore, 304 pages. ISBN 981-238-316-6, 2003.
Errata

Submitted Preprints
* Uniqueness of radial centers of parallel bodies, arXive:1109.5069
* (with R. Langevin, J.C. Sifre) Osculating spheres to a family of curves
* (with K. Mikami and K. Sugahara) Triangles with sides in arithmetic progression

Proceedings without referees in English
* J. O'Hara, Conformal Geometry of Knots, Proceedings of International Conference on "Geometry, Integrable Systems and Visualization", January””26 - 29, 2006, Osaka City University, 62-73
* J. O'Hara, The configuration space of equilateral and equiangular polygons with up to 6 vertices, Bussei Kenkyu Vol.92 no.1 (2009/4) ISSN 0525-2997, 119-122

Preprints
* Criticality for the Knot Energy, MSRI Preprint, 042-94 (1994)
* (with Remi Langevin) Extrinsic Conformal Geometry of Curves and Surfaces, monograph in preparation (always under construction) dt>

Talks given in English
[1] 14 Aug 1990, Energy of a knot, University of Hawaii, Topology Hawaii.
[2] 23 Aug 1990, Energy of a knot, ICM 90 Kyoto.
[3] 31 Aug 1990, Energy of a knot, Post Congress Three Day Symposium on Geometric Topology, University of Tokyo.
[4] 11 June 1991, Energy of a knot, Seminaire Topologie, Universite Pari Sud, Orsay.
[5] 3 Feb 1992, Energy of a knot, Seminar at Max-Planck-Insitut fur Mathematik.
[6] 27 Apr 1994, Energy of a knot, Seminar at Mathematical Sciences Research Institute (MSRI ).
[7] 11 May 1994, Energy of a knot, Topology Seminar at University of California, Davis.
[8] 22 Mar 1996, Energy minimizing knots in $S^3$ and $H^3$, AMS Meeting. Special Session "Physical knot theory", University of Iowa.
[9] 24 Jul 1996, On the existence of the energy minimizing knots, KNOTS' 96, Waseda Universi ty, Tokyo.
[10] 15 Jan 1998, Energy of knots, Conference on Low Dimensional Topology, Universidade da M adeira, Funchal, Portugal.
[11] 11 Aug 1998, Knot energy of higher order, KNOTS' 98 Hellas, European Cultural Center of Delphi, Delphi, Greece.
[12] 20 Sep 1999, Energy of knots, Advanced Course on Integral Geometry, Centre de Recerca M atematica, Universitat Autonoma de Barcelona, Spain.
[13] 13 Oct 1999, Energy of knots, Seminaire general, Universite de Bourgogne, Laboratoire de Topologie, Dijon, France.
[14] 3 Aug 2000, Langevin's conformal invariant knot enery, KNOTS2000, Yongpyong Resort, Korea. Knot 2000. The copies of the OHP transparancies of the lectures are available via web page.
[15] 23 Aug 2000, Langevin's conformal invariant knot enery, IMACS (International Association for Mathematics and Computers in Simulation) quadrennial World Congress on Scientific Computation, Applied Mathematics and Simulation, in Lausanne, Switzerland. IMACS, program and abstract.
[16] 13 Oct 2000, On energy of knots - Energy of knots and Langevin's functional, Spitalfields Day Follow-up: In Search of the Ideal Knot, Isaac Newton Institute for Mathematical Sci ences, Cambridge, U.K.
[17] 04 Jan 2001, Conformal geometry, energy of knots and Langevin's functional, KNOTS, LINKS and MANIFOLDS - 4th International Siegen Topology Symposium - , Siegen, Germany, 4-8 Jan. 2001.
[18] 24 Jan 2001, Conformal geometry and energy functionals for knots, Seminaire General, Laboratoire de Topologie, Universite de Bourgogne, France.
[19] 16 March 2001, Energy of knots and the infinitesimal cross-ratio, Seminaire d'Analyse, EPFL, Lausanne, Switzerland.
[20] 22 April 2001, Energy of knots and the infinitesimal cross-ratio, AMS Sectional Meeting, Univer sity of Nevada, Las Vegas, U.S.A.
[21] 5 June 2001, Energy of knots and the infinitesimal cross-ratio, TOPOLOGICAL FLUID MECHANICS, Cetraro (Calabria, Italy) - June 2-10.
Abstracts of talks
[22] 21 March 2002, Conformally invariant sin $\theta$ energy, Seminaire A.G.T., Laboratoire de Topologie, Universite de Bourgogne, France.
[23] 21 April 2003, The infinitesimal cross ratio and conformally invariant knot energies, S ingularity Theory Seminar, Tokyo Metropolitan University
[24] 3,4 May 2003, The Y-energy of knots, San Francisco, AMS Sprin Western Sectional Meeting .
[25] 18 December 2003, Energy of knots and conformal geometry, Workshop Topology of knots VI", Nihon University, Tokyo, Japan.
[26] 14 October 2004, The space of quasi-equilateral pentagons, Seminaire A.G.T., Institut de Mathematiques de Bourgogne, Universite de Bourgogne, France.
[27] 06 July 2005, Conformal geometry of knots, COE Conference Groups, Homotopy and Config uration Spaces'', The University of Tokyo, Tokyo Japan.
[28] 27 Oct. 2005, Conformal geometry of knots, Singularity and Geometry", Tokyo Universit y of Science, Tokyo Japan.
[29] 1st Nov. 2005, Conformal geometry of curves, Lie Group and Representation Theory Semina r, Kyoto University, RIMS.
[30] 15th Nov. 2005, The infinitesimal cross ratio as an area form, Workshop Extrinsic Con formal Geometry", Stefan Banach International Mathematical Center, Warszawa 10, Poland
[31] 4th Jan. 2006, Energy of knots (survey talk), Workshop Moduli Spaces of Knots", Ameri can Institute of Mathematics, Palo Alto, CA, USA.
[32] 27th Jan. 2006, Conformal geometry of knots, International Conference on Geometry, I ntegrable Systems and Visualization", Osaka CIty University.
Abstract: We study the space S(q,n) of (q+2)-dimensional vector subspaces of the (n+2)-dim ensional Minkowski space which intersect the light cone transversely. It is a subset of an indefinit e Grassmann manifold. This space can be identified with the space of q-spheres in S^n. I will explai n the notion of a pencil, which is one-parameter family of codimension 1 spheres in S^k. Using penci ls, I will give a pseudoorthonormal basis S(q,n).
The pseudo-Riemannian structure of S(q,n) allows us to give an interpretation of the infinitesimal cross ratio", which is a complex valued 2-form on the two point configuraion space of a knot K, KxK -(diagonal): The real part of it can be interpreted as an area element of a surface in S(q,n).
[33] 09 March 2006, Mobius geometry of the set of spheres, International Workshop on Knot T heory for Scientific Objects, Osaka CIty University.
[34] 24 July 2006, The configuration space of a spider, Intelligence of Low Dimensional Top ology 2006, Hiroshima University.
[35] 22 March 2007, The structure of the set of spheres and its applications, Seminaire A.G.T., Institut de Mathematiques de Bourgogne, Universite de Bourgogne, France.
[36] 17 January 2008, The conformal arc-length via osculating circles, Seminaire A.G.T., Institut de Mathematiques de Bourgogne, Universite de Bourgogne, France.
[37] 23 August 2008, Conformal Geometry of curves, 55th Geometry Symposium, Hirosaki University
[38] 29 August 2008, Configuration space of small equilateral and equiangular polygons, Knots and soft-matter physics, Topology of polymers and related topics in physics, mathematics and biology, Yukawa Institute for Theoretical Physics, Kyoto University
[39] 26 May 2009, Conformally invariant energies of knots and links, Advanced School and Conference on Knot Theory and its Applications to Physics and Biology, International Centre for Theoretical Physics, Trieste, Italy.
[40] 3rd Nov. 2009, Introduction to Knot Theory (with Javier Arsuaga), DNA Topology Course 2009, Okinawa Institute of Science and Technology, Okinawa, Japan.
[41] 28 Jul. 2010, Energy of knots and related topics, The 2nd TAPU-KOOK Joint Seminar on Knots and Related Topics & The 4th Graduate Student Workshop on Mathematics, Kyungpook National University, Daegu, Korea.
[42] 3 Aug. 2010, Slopes of double helices and geometric energies, Statistical physics and topology of polymers with ramifications to structure and function of DNA and proteins (2-6, August, 2010, Panasonic Auditorium, Yukawa Hall, YITP, Kyoto University)
[43] 28 June 2011, Renormalization of r*-potentials and generalized centers, Differential Geometry and Parametrization of 3D Knots, Centro di Ricerca Matematica (CRM), Pisa (Italy)
[44] 4 July 2011, M«Óbius invariant energies and average linking with circle, ESF-EMS-CRM-Pi International Conference on Knots and Links: From Form to Function, Centro di Ricerca Matematica (CRM), Scuola Normale Superiore, Pisa
[45] 5 Dec 2012, Renormalized potential energies and their asymptotis, Quantised Flux in Tightly Knotted and Linked Systems, Isaac Newton Institute, Cambridge, UK.
[46] 30 Apr 2013, Three topics in knot energies, Geometric knot theory, MFO Oberwolfach, Germany
[47] 29 Jul 2014, Moebius invariant energies and average linking with circles, A workshop on renormalized energies, Saitama Univ. Japan
[48] 17 March 2015, Energy of knots, regularization and analytic continuation, Seminaris Geometria, Universitat Autonoma de Barcelona, Spain
Invitations (at least one month)
May, June, 1991
Institut des Hautes Etudes Sceintifiques (IHES), France
January - December, 1992
Max-Planck-Institut fur Mathematik, Bonn, Germany.
September, October, 1999
Universite de Bourgogne, Laboratoire de Topologie, Dijon, Franc e (as a guest professor in October). Joint work with Prof. Remi Langevin.
16 August 2000 - 28 March 2001
Universite de Bourgogne, Laboratoire de Topologie, Dijo n, France. Joint work with Prof. Remi Langevin.
6 - 31 March 2002 (three weeks)
Universite de Bourgogne, Laboratoire de Topologie, Dijo n, France. Joint work with Prof. Remi Langevin.
26 August 2002 - 20 October 2002
Universite de Bourgogne, Laboratoire de Topologie, Di jon, France. Joint work with Prof. Remi Langevin.
Activities in abroad (at least one month)
March - May, 1994
Mathematical Sciences Research Institute (MSRI), Berkeley, California, U.S.A., with financial support from the local government of Tokyo.
August 26 2002 - October 20 2002
Universite de Bourgogne, Laboratoire de Topologie, Dij on, France. Joint work with Prof. Remi Langevin.
September 5 2004 - March 29 2005
Universite de Bourgogne, Laboratoire de Topologie, Dijon, France. Joint work with Prof. Remi Langevin.
Supported by Japan Society for the Promotion of Science (JSPS) Scientist Exchanges Bilater al Programs
February 20 - March 31 2007
Universite de Bourgogne, Institut de Mathematiques de Bourgogne, Dijon, France. Joint work with Prof. Remi Langevin.

Grant from the Ministry of educations
1995 Shourei-Kenkyuu A, Low dimensional topology, especially, energy of knots", Number 07 740068
1998-2001, Kiban-Kenkyuu C, Energy of knots (existence of energy minimizers and numerical experiments)", Number 10640085
I return my grant since Aug 15 2000 because of my absense from Japan. The representative of this grant changes to Toshihiko Kurata of our department.
2003-2004, Kiban-Kenkyuu C(2), Energy of knots and conformal geometry", Number 15540088, 1 ,900,000+1,600,000 Yen
2005-2006, Kiban-Kenkyuu C(2), Conformal geometric structure and Knot theory", Number 1754 0089, 1,400,000+1,400,000 Yen
2007-2008, Kiban-Kenkyuu C, Applications of conformal geometry to geometric knot theory", Number 19540096, 700,000+600,000 Yen
2009-2011, Kiban-Kenkyuu C, Conformal Geometry of curves and surfaces and geometric knot theory", Number 21540089, 1,000,000+1,200,000+1,200,00 Yen
ID : 70221132