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March 23, 2026Open Access

Cables of the figure-eight knot via real Frøyshov invariants

Key Points

The aim is to show that the .2n;1/-cable of the figure-eight knot is not smoothly slice when n is odd.
Proved non-smooth slice condition using real Seiberg-Witten Frøyshov invariant.
Developed O.2/-equivariant version of the lattice homotopy type to aid computation.
Computed real Seiberg-Witten Floer homotopy type for specific knots.
Presented calculations of Miyazawa's real framed Seiberg-Witten invariant for 2-knots.
Confirmed that the .2n;1/-cable of the figure-eight knot is not smoothly slice for odd n.
Developed new computational techniques for the real Seiberg-Witten Floer homotopy type.
Provided new insights into the behavior of 2-knots through empirical results.

Abstract

We prove that the .2n;1/-cable of the figure-eight knot is not smoothly slice when n is odd, by using the real Seiberg-Witten Fryshov invariant of Konno, Miyazawa and Taniguchi.For the computation, we develop an O.2/-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 2-knots.

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Kang et al. (Mon,) studied this question.

synapsesocial.com/papers/69c0e007fddb9876e79c16cf https://doi.org/https://doi.org/10.2140/gt.2026.30.781

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