Demonstrates the non-smoothly slice nature of odd cables of the figure-eight knot, suggesting new insights into knot theory.
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. (2026) studied this question.
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