Cables of the figure-eight knot via real Fr{\o}yshov invariants

TL;DR

This paper demonstrates that certain cables of the figure-eight knot are not smoothly slice using real Seiberg-Witten invariants, and introduces new computational methods for these invariants.

Contribution

It develops an $O(2)$-equivariant lattice homotopy type to compute real Seiberg-Witten Floer homotopy types for knots, advancing the computational tools in knot theory.

Findings

$(2n,1)$-cable of the figure-eight knot is not smoothly slice for odd n

Introduces an $O(2)$-equivariant lattice homotopy type for computations

Provides new calculations of Miyazawa's real framed Seiberg-Witten invariant for 2-knots

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 Fr{\o}yshov 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.