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Smooth concordance of cables of the figure-eight knot

Sungkyung Kang, JungHwan Park, Masaki Taniguchi

TL;DR

This work resolves the smooth sliceness problem for all cables of the figure-eight knot by introducing higher-order real concordance invariants $\kappa_R^{(k)}$, constructed from $2^k$-fold branched covers and real Seiberg--Witten Floer $K$-theory. The authors develop a uniform framework applicable to all $(2^k m,1)$-cables, establish a real 10/8-type inequality for these invariants, and compute them for torus knots via two complementary methods (MOY96 and DSS2023), showing that nontrivial cables have infinite order in the smooth concordance group. The topology of Brieskorn spheres and AR plumbing graphs, together with Levine--Tristram signatures, underpins the calculations, while the $\mathbb{Z}_4$-equivariant stable-homotopy setting provides the structural foundation for the higher invariants. The results yield new torsion-related phenomena in concordance and extend obstructions beyond existing involutive Floer techniques, with potential implications for Floer-thin knot families and related four-manifold topology.

Abstract

We prove that every nontrivial cable of the figure-eight knot has infinite order in the smooth knot concordance group. Our main contribution is a uniform proof that applies to all $(2n,1)$-cables of the figure-eight knot. To this end, we introduce a family of concordance invariants $κ_R^{(k)}$, defined via $2^k$-fold branched covers and real Seiberg--Witten Floer $K$-theory. These invariants generalize the real $K$-theoretic Frøyshov invariant developed by Konno, Miyazawa, and Taniguchi.

Smooth concordance of cables of the figure-eight knot

TL;DR

This work resolves the smooth sliceness problem for all cables of the figure-eight knot by introducing higher-order real concordance invariants , constructed from -fold branched covers and real Seiberg--Witten Floer -theory. The authors develop a uniform framework applicable to all -cables, establish a real 10/8-type inequality for these invariants, and compute them for torus knots via two complementary methods (MOY96 and DSS2023), showing that nontrivial cables have infinite order in the smooth concordance group. The topology of Brieskorn spheres and AR plumbing graphs, together with Levine--Tristram signatures, underpins the calculations, while the -equivariant stable-homotopy setting provides the structural foundation for the higher invariants. The results yield new torsion-related phenomena in concordance and extend obstructions beyond existing involutive Floer techniques, with potential implications for Floer-thin knot families and related four-manifold topology.

Abstract

We prove that every nontrivial cable of the figure-eight knot has infinite order in the smooth knot concordance group. Our main contribution is a uniform proof that applies to all -cables of the figure-eight knot. To this end, we introduce a family of concordance invariants , defined via -fold branched covers and real Seiberg--Witten Floer -theory. These invariants generalize the real -theoretic Frøyshov invariant developed by Konno, Miyazawa, and Taniguchi.
Paper Structure (15 sections, 17 theorems, 110 equations, 3 figures)

This paper contains 15 sections, 17 theorems, 110 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Previous Work
  3. Our contribution
  4. Some topological computations
  5. An AR negative-definite plumbing graph for $\Sigma(2^k,2^k \cdot m,2^k \cdot 10m - 1)$
  6. On Brieskorn spheres of the form $\Sigma(2^k,2^k\cdot m,q)$
  7. Levine-Tristram signatures of torus knots $T_{pm,pmn-1}$
  8. Branched covers along the disks $D_{k,m}$
  9. Concordance invariants from higher covers
  10. Notations
  11. The higher real concordance invariants and inequalities
  12. The computation for torus knots
  13. Computation via Mrowka--Ozsváth--Yu
  14. Computation via Dai-Sasahira-Stoffregen
  15. Proof of the main theorem

Key Result

Theorem 1.1

Every nontrivial cable of the figure-eight knot has infinite order in the smooth knot concordance group.

Figures (3)

  • Figure 1: A surgery diagram for $\partial W_{\Gamma_{k,m,q}}$. The action of $\tau$, a generator of $\mathbb{Z}_{2^k}$, is given by rotation by $\frac{2\pi}{2^k}$ about the red vertical surgery curve.
  • Figure 2: A surgery diagram for the quotient 3-manifold $\partial W_{\Gamma_{k,m,q}}/\mathbb{Z}_{2^k}$. The branching set $K$, namely the image of $\mathrm{Fix}(\tau)$ under the projection $\partial W_{\Gamma_{k,m,q}} \to \partial W_{\Gamma_{k,m,q}}/\mathbb{Z}_{2^k}$, is drawn in red.
  • Figure 3: A smooth concordance $S$ in the twice-punctured $2\mathbb{CP}^2$ from the figure-eight knot to the unknot.

Theorems & Definitions (35)

  • Theorem 1.1
  • Lemma 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • Lemma 2.7
  • ...and 25 more