Cables of the figure-eight knot via real Frøyshov invariants

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 Frøyshov 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.

Revised version; 39 pages, several figures

References in corpus (7)

• Chern-Simons functional, singular instantons, and the four-dimensional clasp number
• Equivariant Seiberg-Witten-Floer cohomology
• Almost simple linear graphs, homology cobordism and connected Heegaard Floer homology
• Knot concordance invariants from Seiberg-Witten theory and slice genus bounds in 4-manifolds
• A note on the four-dimensional clasp number of knots
• Instantons, special cycles, and knot concordance
• Involutions, knots, and Floer K-theory