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Iterated satellite operators on the knot concordance group

Jae Choon Cha, Taehee Kim

TL;DR

This work analyzes iterated satellite operators on the knot concordance group 𝒞, focusing on winding number zero patterns with axis η whose Blanchfield self-pairing is nontrivial. By combining higher-order amenable L^2-signatures with Blanchfield-bordism technology, the authors show that every graded piece ⟨P^n(𝒞)⟩/⟨P^{n+1}(𝒞)⟩ has infinite rank for all n, and that each iteration P^n is not a homomorphism unless the operator is trivial or an orientation reversal. The results extend prior work on robust doubling operators, apply in both topological and smooth settings, and yield new distinctness results when comparing different patterns with relatively prime Alexander polynomials. The methods hinge on constructing tailored 4-manifolds and using the amenable signature theorem to force linear independence via nonzero rho-invariants across levels of the P-filtration, and they provide partial answers to questions of Hedden and Pinzón-Caicedo regarding infinite-rank behavior in iterated satellites.

Abstract

We show that for a winding number zero satellite operator $P$ on the knot concordance group, if the axis of $P$ has nontrivial self-pairing under the Blanchfield form of the pattern, then the image of the iteration $P^n$ generates an infinite rank subgroup for each $n$. Furthermore, the graded quotients of the filtration of the knot concordance group associated with $P$ have infinite rank at all levels. This gives an affirmative answer to a question of Hedden and Pinzón-Caicedo in many cases. We also show that under the same hypotheses, $P^n$ is not a homomorphism on the knot concordance group for each $n$. We use amenable $L^2$-signatures to prove these results.

Iterated satellite operators on the knot concordance group

TL;DR

This work analyzes iterated satellite operators on the knot concordance group 𝒞, focusing on winding number zero patterns with axis η whose Blanchfield self-pairing is nontrivial. By combining higher-order amenable L^2-signatures with Blanchfield-bordism technology, the authors show that every graded piece ⟨P^n(𝒞)⟩/⟨P^{n+1}(𝒞)⟩ has infinite rank for all n, and that each iteration P^n is not a homomorphism unless the operator is trivial or an orientation reversal. The results extend prior work on robust doubling operators, apply in both topological and smooth settings, and yield new distinctness results when comparing different patterns with relatively prime Alexander polynomials. The methods hinge on constructing tailored 4-manifolds and using the amenable signature theorem to force linear independence via nonzero rho-invariants across levels of the P-filtration, and they provide partial answers to questions of Hedden and Pinzón-Caicedo regarding infinite-rank behavior in iterated satellites.

Abstract

We show that for a winding number zero satellite operator on the knot concordance group, if the axis of has nontrivial self-pairing under the Blanchfield form of the pattern, then the image of the iteration generates an infinite rank subgroup for each . Furthermore, the graded quotients of the filtration of the knot concordance group associated with have infinite rank at all levels. This gives an affirmative answer to a question of Hedden and Pinzón-Caicedo in many cases. We also show that under the same hypotheses, is not a homomorphism on the knot concordance group for each . We use amenable -signatures to prove these results.
Paper Structure (16 sections, 34 theorems, 112 equations, 4 figures)

This paper contains 16 sections, 34 theorems, 112 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Graded quotients and classical signature invariants
  3. Preliminaries on L2-signature obstructions
  4. Iterated satellite operators of infinite rank
  5. Proof of Theorem \ref{['theorem:main']}
  6. Proof of Proposition \ref{['proposition:computation-of-rho-invariant-of-J_0^i']}
  7. Lagrangian bordism and Blanchfield bordism
  8. Satellite operators that are not homomorphisms
  9. Proof of Theorem \ref{['theorem:non-homomorphism']}
  10. Proof of Proposition \ref{['proposition:computation-of-rho(J)-non-homomorphism']}
  11. Satellites of patterns with relatively prime Alexander polynomials
  12. Proof of Theorem \ref{['theorem:coprime-satellites']}
  13. Proof of Proposition \ref{['proposition:phi-coprime']}
  14. Slice satellite operators
  15. A special case of Theorem \ref{['theorem:main']} for slice satellite operators
  16. ...and 1 more sections

Key Result

Theorem 1

If a satellite operator $P\colon \mathcal{C}\longrightarrow \mathcal{C}$ has winding number zero and $\mathop{B\ell}(\eta,\eta)$ is nontrivial for the axis $\eta$, then $\langle P^n(\mathcal{C})\rangle/\langle P^{n+1}(\mathcal{C})\rangle$ has infinite rank for all $n\ge 0$. In particular, $\langle P

Figures (4)

  • Figure 1: A satellite knot $P(K)$ with pattern $P$ and companion $K$.
  • Figure 2: The cobordism $W_0$
  • Figure 3: The cobordism $W_0$.
  • Figure 4: The cobordism $W_0$

Theorems & Definitions (72)

  • Conjecture 1: Hedden-Pinzon-Caicedo:2021-1
  • Theorem 1
  • Remark 1.1: A question on topologically slice iterated satellite knots
  • Proposition 1.2
  • Proposition 1.3
  • Conjecture 3: Hedden
  • Theorem 2
  • Theorem 3
  • Lemma 2.1: Cha-Livingston:2002-1
  • proof : Proof of Proposition \ref{['proposition:zeroth-quotient']}
  • ...and 62 more