Research

• Knot energies (Wiki)  (a figure illustrating the motivation), Möbius energy (Wiki), one hour presentation file, movies of knots evolving to decrease the energy are available on web, by Kazushi Ahara  and by John Sullivan. See also Knotplot cite (illustrated in Sketching a knot using KnotPlot) by Rob Scharein.

• Möbius Geometry of curves and surfaces, one hour presentation file

• Regularization of Riesz potential and energy of submanifolds and generalized centers, one hour presentation file

• Knot space (Configuration space of equilateral and equiangular hexagons)

• Optimal museum gallery problem, Program (change .eye to .exe) and instruction manual (ppt) (in Japanese) by Akio Fukuda.

Research abstract in 4 pages (Dec. 2018)

My References, References of Möbius Geometry (Feb 2020)

1. Knot energies.

Key Words: Coulomb's force, electrostatic energy, energy functional of knots, ideal knot, numerical experiments, energy minimizer, prime knots, Möbius invariance.

1. Ideal, best packing, and energy minimizing double helices, Progress of Theoretical Physics Supplement 191 (2011) 215-224, arXiv:1104.0489, one hour presentation file

2. 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.

3. 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. arXive.

4. Asymptotic behavior 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

5. Energy of knots, "Ideal Knots", A. Stasiak, V. Katrich, L. H. Kauffman eds., World Scientific, 1998, 288-314

6. Energy of knots, Sugaku Expositions 13 ,2000, 73-90

7. 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

8. Energy funcitonals of knots II, Topology Appl. 56 (1994), 45-61

9. Energy functionals of knots, in "Topology -Hawaii", K.H. Dovermann ed., World Scientific, Singapore, (1992), 201-214

10.  Family of energy functionals of knots, Topology Appl. 48 (1992), 147-161

11. Energy of a knot, Topology 30 (1991), 241-247     

• Möbius invariant metrics on the space of knots, to appear in Geom. Dedicata, DOI: 10.1007/s10711-020-00518-6 arXiv 1905.06098

• Regularization of Neumann and Weber formulae for inductance, Journal of Geometry and Physics 149 (2020), 10.1016/j.geomphys.2019.103567, arXiv 1804.11098

are also related to knot energies

2. Möbius Geometry of curves and surfaces.

Key Words: Möbius transformation, inversion in a sphere, Minkowski space, Lorentz metric, pencil of spheres, cross ratio, infinitesimal cross ratio, Plucker coordinates, osculating circle, canonical symplectic form of a cotangent bundle, Kahler form.

1. Möbius invariant metrics on the space of knots, to appear in Geom. Dedicata, DOI: 10.1007/s10711-020-00518-6 arXiv 1905.06098

2. (with Gil Solanes) Möbius invariant energies and average linking with circles, Tohoku Math. J. 67 (2015),   51-82 arXiv:1010.3764, one hour presentation file

3. (with H. Funaba) Möbius invariant energy of tori of revolution, Journal of Physics: Conference Series, Volume 544, Issue 1, article id. 012019 (2014) arXiv:1305.3682

4. Measure of a 2-component link, Tohoku Math. J. Tohoku Math. J. 65 (2013), 427-440 arXiv:0709.2215

5. (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 arXiv:1102.0344

6. (with Udo Hertrich-Jeromin and Alastair King) On the Moebius geometry of Euclidean triangles, Elemente der Mathematik 68 (2013), 96-114.

7. (with Remi Langevin) Conformal invariance of the writhe of a knot, J. Knot Theory Ramifications 19 (2010) 1115-1123, arXiv:0803.1876

8. (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

9. (with R. Langevin) Conformally invariant energies of knots, J. Institut Math. Jussieu 4 (2005), 219-280. arXive, abstract, one hour presentation file

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.

3. Energies and residues of submanifolds and generalized centers.

Key Words: Riesz potential, Riesz energy, regularization, Hadamard's finite part, analytic continuation, Brylinski beta function, residue, convex geometry, integral geometry, dual mixed volume, radial center, minimal unfolded region, heart, intrinsic volume, Quermassintegrale.

1. (with Gil Solanes) Erratum to Regularized Riesz energies of submanifolds, to appear in Math. Nachr. DOI: 10.1002/mana.202000024

2. Regularization of Neumann and Weber formulae for inductance, Journal of Geometry and Physics 149 (2020), 10.1016/j.geomphys.2019.103567, arXiv 1804.11098

3. Characterization of balls by generalized Riesz energy, Math. Nachr. 292 (2019), 159-169, DOI: 10.1002/mana.201700256,  arXiv:1707.02405

4. (with Gil Solanes)  Regularized Riesz energies of submanifolds, Math. Nachr. 291 (2018), 1356-1373, DOI: 10.1002/mana.201600083 , arXiv:1512.07935, one hour presentation file

5. Minimal unfolded regions of a convex hull and parallel bodies, Hokkaido Math. J.  44 (2015), 175-183  arXiv:1205.0662

6. Renormalization of potentials and generalized centers, Adv. Appl. Math. 48 (2012), 365-392, arXiv:1008.2731 ;         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)