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Decomposition of generalized O’Hara’s energies

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Abstract

O’Hara introduced several functionals as knot energies. One of them is the Möbius energy. We know its Möbius invariance from Doyle-Schramm’s cosine formula. It is also known that the Möbius energy was decomposed into three components keeping the Möbius invariance. The first component of decomposition represents the extent of bending of the curves or knots, while the second one indicates the extent of twisting. The third one is an absolute constant. In this paper, we show a similar decomposition for generalized O’Hara energies. We also extend the cosine formula for the Möbius energy to generalized O’Hara energies. It gives us a condition for which the right circle minimizes the energy under the length-constraint. Furthermore, it shows us how far the energy is from the Möbius invariant property. Using decomposition, the first and second variation formulae are derived.

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Acknowledgements

The authors express their appreciation to Professor Jun O’Hara for providing access to [10].

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Authors and Affiliations

  1. Department of Mathematics, Graduate School of Science and Engineering, Saitama University, Shimo-Okubo 255, Sakura-ku, Saitama-shi, Saitama, 338-8570, Japan

    Aya Ishizeki & Takeyuki Nagasawa

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  1. Aya Ishizeki
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  2. Takeyuki Nagasawa
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Correspondence to Aya Ishizeki.

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The Aya Ishizeki is supported by Grant-in-Aid for JSPS Fellows (No. 17J01429) and the Takeyuki Nagasawa is supported by Grant-in-Aid for Scientific Research (C) (No. 17K05310), Japan Society for Promotion of Science.

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Ishizeki, A., Nagasawa, T. Decomposition of generalized O’Hara’s energies. Math. Z. 298, 1049–1076 (2021). https://doi.org/10.1007/s00209-020-02601-w

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  • DOI: https://doi.org/10.1007/s00209-020-02601-w

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