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Cables of the figure-eight knot via real Frøyshov invariants

Sungkyung Kang, JungHwan Park, Masaki Taniguchi

TL;DR

This work shows that the $(2n,1)$-cable of the figure-eight knot is not smoothly slice for odd $n$ by leveraging real Frøyshov invariants derived from the real Seiberg–Witten Floer homotopy type $SWF_R(K)$. The authors introduce an $O(2)$-equivariant lattice homotopy framework to compute $SWF_R(K)$ for almost rational plumbed knots, and they derive explicit invariants for torus knots $T_{2n,1-20n}$ that feed into a concordance-based obstruction. By transporting lattice data to $SWF_R$ via almost $I$-invariant paths and constructing $O(2)$-equivariant Bauer–Furuta maps, they obtain concrete obstructions that imply $c_4^+(E_{2n,1})>0$ for odd $n$. The approach extends to arborescent and Montesinos-type knots, yielding a broader obstruction toolkit and a structural view of $O(2)$-equivariant Bauer–Furuta invariants in relation to Miyazawa’s degree 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.

Cables of the figure-eight knot via real Frøyshov invariants

TL;DR

This work shows that the -cable of the figure-eight knot is not smoothly slice for odd by leveraging real Frøyshov invariants derived from the real Seiberg–Witten Floer homotopy type . The authors introduce an -equivariant lattice homotopy framework to compute for almost rational plumbed knots, and they derive explicit invariants for torus knots that feed into a concordance-based obstruction. By transporting lattice data to via almost -invariant paths and constructing -equivariant Bauer–Furuta maps, they obtain concrete obstructions that imply for odd . The approach extends to arborescent and Montesinos-type knots, yielding a broader obstruction toolkit and a structural view of -equivariant Bauer–Furuta invariants in relation to Miyazawa’s degree invariants.

Abstract

We prove that the -cable of the figure-eight knot is not smoothly slice when is odd, by using the real Seiberg-Witten Frøyshov invariant of Konno-Miyazawa-Taniguchi. For the computation, we develop an -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.
Paper Structure (28 sections, 27 theorems, 226 equations, 4 figures)

This paper contains 28 sections, 27 theorems, 226 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Some topological facts
  3. Concordance to torus knots
  4. Topological invariants for torus knots
  5. Review of the $\delta_R$ invariant and the case $n=1$
  6. Category of spectrums for $SWF_R(K)$
  7. The real Frø yshov invariants
  8. $O(2)$-equivariant Floer homotopy type
  9. $O(2)$-equivariant cobordism map
  10. The case $n=1$: a toy model
  11. Two technical lemmas
  12. Lattice homotopy type and the proof of \ref{['thm:main']}
  13. Computation sequences in lattice homology
  14. Review of $S^1$ and $\mathrm{Pin}(2)$-lattice homotopy type
  15. Involutions on plumbed 4-manifolds and almost $I$-invariant paths
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $E$ be the figure-eight knot, and let $E_{2n,1}$ denote the $(2n,1)$-cable of $E$. For each positive odd integer $n$, the knot $E_{2n,1}$ does not bound a normally immersed disk in $B^4$ with only negative double points. In particular, for each odd integer $n$, the knot $E_{2n,1}$ is not smoothl

Figures (4)

  • Figure 1: The $0$-framed figure-eight knot becomes the $-10$-framed unknot after two full negative twists.
  • Figure 2: A surgery diagram for $\partial W_{\Gamma_{p,q}}$. The action of $\tau$ can be seen as the $180^\circ$ rotation about the red vertical surgery curve. Here, $\beta_2$ and $\beta_3$ are negative integers satisfying the equation $pq + q\beta_2 + p\beta_3 = 1.$
  • Figure 3: A surgery diagram of $\partial W_{\Gamma_{p,q}}/\tau$. The branching set $K$, namely the image of $\mathrm{Fix}(\tau)$ under the projection $\partial W_{\Gamma_{p,q}} \to \partial W_{\Gamma_{p,q}}/\tau$, is drawn in red.
  • Figure 4: A surgery diagram for the $(pq+N)$-surgery along $K$.

Theorems & Definitions (57)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Remark 1.8
  • Proposition 2.1
  • Remark 2.2
  • ...and 47 more