Abstract
We give a condition for a function to produce a Möbius invariant weighted inner product on the tangent space of the space of knots, and show that some kind of Möbius invariant knot energies can produce Möbius invariant and parametrization invariant weighted inner products. They would give a natural way to study the evolution of knots in the framework of Möbius geometry.
Similar content being viewed by others
References
Banchoff, T., White, J.H.: The behavior of the total twist and self-linking number of a closed space curve under inversions. Math. Scand. 36, 254–262 (1975)
Blatt, S.: The gradient flow of the Möbius energy near local minimizers. Calc. Var. Partial Differ. Equ. 43, 403–439 (2012). https://doi.org/10.1007/s00526-011-0416-9
Blatt, S., Reiter, P., Schikorra, A.: Harmonic analysis meets critical knots. Critical points of the Möbius energy are smooth. Trans. Am. Math. Soc. 368, 6391–6438 (2016)
Blatt, S., Vorderobermeier, N.: On the analyticity of critical points of the Möbius energy. preprint, arXiv:1805.05739
Brylinski, J.-L.: The beta function of a knot. Int. J. Math. 10, 415–423 (1999)
Fialkow, A.: The conformal theory of curves. Trans. Am. Math. Soc. 51, 435–501 (1942)
Freedman, M.H., He, Z.-X., Wang, Z.: Möbius energy of knots and unknots. Ann. Math 139, 1–50 (1994)
Gilsbach, A., von der Mosel, H.: Symmetric critical knots for O’Hara’s energies. Topol. Appl. 242, 73–102 (2018)
He, Z.-X.: The Euler–Lagrange equation and heat flow for the Möbius energy. Commun. Pure Appl. Math. 53, 399–431 (2000)
Ishizeki, A., Nagasawa, T.: A decomposition theorem of the Möbius energy I: decomposition and Möbius invariance. Kodai Math. J. 37, 737–754 (2014)
Ishizeki, A., Nagasawa, T.: A decomposition theorem of the Möbius energy II: variational formulae and estimates. Math. Ann. 363, 617–635 (2015)
Kusner, R., Sullivan, J.M.: Möbius-invariant Knot Energies. In: Stasiak, A., Katrich, V., Kauffman, L.H. (eds.) Ideal Knots, pp. 315–352. World Scientific, Singapore (1998)
Langevin, R., O’Hara, J.: Conformally invariant energies of knots. J. Inst. Math. Jussieu 4, 219–280 (2005)
Nakauchi, N.: A remark on O’Hara’s energy of knots. Proc. Am. Math. Soc. 118, 293–296 (1993)
O’Hara, J.: Energy of a knot. Topology 30, 241–247 (1991)
O’Hara, J., Solanes, G.: Möbius invariant energies and average linking with circles. Tohoku Math. J. 67, 51–82 (2015)
O’Hara, J., Solanes, G.: Regularized Riesz energies of submanifolds. Math. Nachr. 291, 1356–1373 (2018)
Reiter, P.: Repulsive knot energies and pseudodifferential calculus for O’Hara’s knot energy family \(E(\alpha )\)\(\alpha \in [2,3)\). Math. Nachr. 285, 889–913 (2012)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supported by JSPS KAKENHI Grant No. 16K05136 and 19K03462.
Rights and permissions
About this article
Cite this article
O’Hara, J. Möbius invariant metrics on the space of knots. Geom Dedicata 209, 1–13 (2020). https://doi.org/10.1007/s10711-020-00518-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10711-020-00518-6