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The Dabkowski-Sahi invariant and 4-moves for links

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Abstract

Dabkowski and Sahi defined an invariant of a link in the 3-sphere, which is preserved under 4-moves. This invariant is a quotient of the fundamental group of the complement of the link. It is generally difficult to distinguish between the Dabkowski-Sahi invariants of given links. In this paper, we give a necessary condition for the existence of an isomorphism between the Dabkowski-Sahi invariant of a link and that of the corresponding trivial link. Using this condition, we provide a practical obstruction to a link to be trivial up to 4-moves.

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Notes

  1. Since n and q can be chosen arbitrarily large, the condition \(j<\min \{n,q\}\) is no real restriction.

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Acknowledgements

The authors would like to thank the referee for valuable comments on an early version of this paper, and in particular for suggesting us to work on 3-component links, which led to Theorem 3.4 and Proposition 4.4. The second author was supported by JSPS KAKENHI Grant Number JP21K20327. The third author was supported by JSPS KAKENHI Grant Number JP21K03237 and Waseda University Grant for Special Research Projects (Project number: 2021C-120).

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Authors and Affiliations

  1. Institute for Mathematics and Computer Science, Tsuda University, 2-1-1 Tsuda-machi, Kodaira, Tokyo, 187-8577, Japan

    Haruko A. Miyazawa

  2. Department of Mathematics, Kobe University, 1-1 Rokkodai, Nada-ku, Kobe, 657-8501, Japan

    Kodai Wada

  3. Faculty of Commerce, Waseda University, 1-6-1 Nishi-Waseda, Shinjuku-ku, Tokyo, 169-8050, Japan

    Akira Yasuhara

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  1. Haruko A. Miyazawa
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Correspondence to Kodai Wada.

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Miyazawa, H.A., Wada, K. & Yasuhara, A. The Dabkowski-Sahi invariant and 4-moves for links. Geom Dedicata 217, 46 (2023). https://doi.org/10.1007/s10711-023-00780-4

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