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Cables of the
figure-eight knot via real Frøyshov invariants
Sungkyung Kang, JungHwan Park and Masaki Taniguchi |
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Geometry & Topology 30 (2026) 781–833
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Abstract |
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We prove that the -cable of the figure-eight knot is not smoothly slice when is odd, by using the real Seiberg–Witten Frøyshov invariant of Konno, Miyazawa and Taniguchi. For the computation, we develop an -equivariant version of the lattice homotopy type, originally introduced by Dai, Sasahira and Stoffregen. This enables us to compute the real Seiberg–Witten Floer homotopy type for a certain class of knots. Additionally, we present some computations of Miyazawa’s real framed Seiberg–Witten invariant for -knots. |
Keywords
slice-ribbon conjecture, real Frøyshov invariant,
figure-eight knot, Miyazawa's invariant
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Mathematical Subject Classification
Primary: 57K10, 57K41
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References |
Publication
Received: 30 November 2024
Revised: 27 June 2025
Accepted: 28 July 2025
Published: 16 March 2026
Proposed: Ciprian Manolescu
Seconded: András I Stipsicz, Peter Ozsváth
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Authors |
| Sungkyung Kang | |
| Mathematical Institute University of Oxford Oxford United Kingdom |
Department of Pure Mathematics and
Mathematical Statistics University of Cambridge Cambridge United Kingdom |
| JungHwan Park | |
| Department of Mathematical
Sciences KAIST Daejeon South Korea |
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| Masaki Taniguchi | |
| Department of Mathematics Kyoto University Kyoto Japan |
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| © 2026 The Author(s), under exclusive license to MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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