Cables of the figure-eight knot via real Fryshov invariants

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-Taniguchi. For the computation, we develop an O (2) -equivariant version of the lattice homotopy type, originally introduced by Dai-Sasahira-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.