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Filtered Instantons and the Concordance of Satellites

Ivan So

TL;DR

This work uses the $SU(2)$-instanton-based filtered Floer invariant $r_s$ to study satellite operations in the smooth knot concordance group. By deriving a surgery description for branched double covers via the Montesinos trick and rational tangles, the authors connect satellite patterns to $S^3_{1/n}$-surgeries on connected sums, enabling $r_s$-based obstructions to linear independence. A practical corollary employs the slice torus invariant \\widetilde{s}$ to certify negativity of $h(S^3_1(K\\# J\\# K))$, yielding broad linear independence results for families like the Whitehead $n$-ble $P_n$ and other patterns, sometimes with explicit nontrivial Montesinos friends. The paper also contrasts the $r_s$ approach with prior $SO(3)$-instanton methods, noting greater computability in certain cases and providing explicit examples demonstrating independence beyond torus companions. Overall, the results advance understanding of how satellite operations act on the smooth concordance group and illustrate the power of filtered instanton techniques in producing concrete linear-independence criteria.

Abstract

In \cite{NST23}, Nozaki-Sato-Taniguchi defined a family of invariants $ r_s $ for integer homology spheres with filtered instanton homology \cite{FS92}. Coupling these with techniques in classical knot theory, we produce some results in the knot concordance group, including criteria for a family of satellite knots to be linearly independent and the independence of what we call the Whitehead $ n $-ble $ P_n $.

Filtered Instantons and the Concordance of Satellites

TL;DR

This work uses the -instanton-based filtered Floer invariant to study satellite operations in the smooth knot concordance group. By deriving a surgery description for branched double covers via the Montesinos trick and rational tangles, the authors connect satellite patterns to -surgeries on connected sums, enabling -based obstructions to linear independence. A practical corollary employs the slice torus invariant \\widetilde{s}h(S^3_1(K\\# J\\# K))nP_nr_sSO(3)$-instanton methods, noting greater computability in certain cases and providing explicit examples demonstrating independence beyond torus companions. Overall, the results advance understanding of how satellite operations act on the smooth concordance group and illustrate the power of filtered instanton techniques in producing concrete linear-independence criteria.

Abstract

In \cite{NST23}, Nozaki-Sato-Taniguchi defined a family of invariants for integer homology spheres with filtered instanton homology \cite{FS92}. Coupling these with techniques in classical knot theory, we produce some results in the knot concordance group, including criteria for a family of satellite knots to be linearly independent and the independence of what we call the Whitehead -ble .

Paper Structure

This paper contains 16 sections, 19 theorems, 20 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Independence in the concordance group
  3. Linearly independence under fixed Whitehead $n$-ble
  4. Comparison to prior technologies
  5. Surgery Description of Branched Double Cover
  6. Rational tangles
  7. The Montesinos trick for patterns of rational unknotting number one
  8. Surgery description of $\Sigma_2(P_n(K))$
  9. Filtered instanton Floer homology and the instantons filtration invariant
  10. Filtered instantons and the definition of $r_s$
  11. Properties of $r_s$
  12. Concordance invariant $\widetilde{s}$ from special cycles
  13. Proof of the main theorems
  14. Proof of the linear independence
  15. Independence under Whitehead $n$-ble
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $Q_n\subset S^1\times D^2$ be a family of patterns such that where $h:\Theta^3_\mathbb{Z}\to\mathbb{Z}$ is the Frø yshov homomorphism from Fro02, then $\{Q_n(K)\}_{n\in\mathbb{N}}$ are linearly independent in $\mathcal{C}$.

Figures (8)

  • Figure 1: The pattern $P_n$ considered in this paper. Here $n$ denote the number of positive half-twist.
  • Figure 2: Example of rational tangles.
  • Figure 3: Realization of $J$.
  • Figure 4: Computation of the surgery coefficient.
  • Figure 5: Determination of $J$ for the pattern $P_n$.
  • ...and 3 more figures

Theorems & Definitions (27)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Definition 1.5
  • Theorem 1.6: HK12, Theorem 1
  • Theorem 1.7
  • Remark 1.8
  • Definition 2.1
  • Example 2.2
  • ...and 17 more