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The crossing numbers of amphicheiral knots

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Abstract

We determine the crossing numbers of (prime) amphicheiral knots. This problem dates back to the origin of knot tables by Tait and Little at the end of the nineteenth century. The proof is the most substantial application of the semiadequacy formulas for the edge coefficients of the Jones polynomial.

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Data availability

Programs used for the most substantial computations are available at [45]. The data obtained during the computations are explained in the text. Further details of the procedures involved can be discussed upon request.

Notes

  1. Beware in particular of confusing the words ‘loop’ and ‘cycle’; the meanings they are used in here are very different! A loop is a piece of a diagram obtained after splicing all crossings, and a cycle is a set of loops connected by splicing traces in an appropriate manner.

  2. Note, though, that \(a_A(D)\) is called m(D) in [40], and a(D) is used there with a different meaning.

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Acknowledgements

Initial part of this research was supported by a JSPS fellowship (I would wish to thank to Prof. T. Kohno at U. Tokyo for his hospitality) and by the Japan 21st Century COE Program. The work was further partially funded by the National Research Foundation of Korea through the Korean Ministry of Science and ICT (MSIT, grants NRF-2017R1E1A1A03071032 and 2023R1A2C1003749) and the International Research & Development Program of MSIT (grant NRF-2016K1A3A7A03950702). The referee made some helpful suggestions.

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  1. Department of Mathematics Education, Dongguk University, WISE Campus, 123, Dongdae-ro, Gyeongju-si, 38066, Republic of Korea

    A. Stoimenow

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Stoimenow, A. The crossing numbers of amphicheiral knots. Res Math Sci 11, 34 (2024). https://doi.org/10.1007/s40687-024-00440-3

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