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Higher-order rho bounds and coprime separation

Statement

Assume full AC and a smooth winding-zero pattern $P$ whose axis has nonzero integral classical Blanchfield self-value. Use oriented zero surgery $M (K)$ and the first-positive-push Levine–Tristram signature function $σ_{J}$ . Let $L > 0$ exceed the uniform Cheeger–Gromov bound for every rho value of $M (P (U))$ under every group representation. For every $n, m \geq 1$ , there are distinct odd primes $p_{1}, \dots, p_{m}$ outside the finite exceptional set for reduction of this self-value and finite disk-band companion knots $J_{i}$ such that, writing $ζ_{p} = \exp (2 π i / p)$ , $D_{i} = \frac{1}{p_{i}} \sum_{r = 0}^{p_{i} - 1} σ_{J_{i}} (ζ_{p_{i}}^{r}) > m n L,$ with $σ_{J_{i}} (1) = 0$ , $σ_{J_{i}} (ζ_{p_{i}}) = σ_{J_{i}} (ζ_{p_{i}}^{- 1}) > p_{i} m n L / 2$ , all other nonidentity $p_{i}$ -th-root signatures zero, and every $p_{j}$ -th-root signature zero for $j \neq i$ . Thus $D_{i}$ is the canonical $Z / p_{i}$ cyclic defect of $M (J_{i})$ . In any candidate smooth bounding bordism for a nonzero integer vector $(a_{i})$ used in the actual finite-prime infection obstruction assembly, choose $i_{*}$ with $∣ a_{i_{*}} ∣ = \max_{i} ∣ a_{i} ∣$ . Its supplied quotient using $p_{i_{*}}$ gives signed bottom boundary contribution $a_{i_{*}} D_{i_{*}}$ ; the remaining $n \sum_{i} ∣ a_{i} ∣$ pattern terms have total absolute value less than $n L \sum_{i} ∣ a_{i} ∣ \leq n m L ∣ a_{i_{*}} ∣ < ∣ a_{i_{*}} ∣ D_{i_{*}}$ . Thus the total signed boundary rho value has the sign of $a_{i_{*}}$ and is nonzero.

For every $n \geq 1$ , the three-branch obstruction assembly also admits a finite disk-band companion knot with a canonical cyclic defect $D > 3 n L$ . For any supplied candidate smooth homology-circle bound in that assembly, its quotient kills the distinguished negative bottom branch and gives either $0$ or $D$ on each positive bottom branch, with at least one value $D$ . Its $3 n$ remaining pattern terms have total absolute value less than $3 n L$ , so the total boundary rho value is positive. For a locally flat TOP pair-family application, the initial bordism must supply the actual height- $n$ pairs and height- $(n + 1)$ isotropic lifts; for a TOP three-branch application it must supply the actual homology-circle bound. In either case the finite-projective coefficient quotient must satisfy the stated survival and vanishing hypotheses, and the boundary rho values must agree with the smooth bounding defects by the closed TOP bordism equality. Pairwise relatively prime Alexander polynomials belong only to the separate Theorem C comparison of distinct patterns; they are not hypotheses of Theorems A or B.

Statement references

Cha–Kim, printed3–5,9–14,19–25

Cha–Livingston, Theorem1

Facts & Assumptions

Given: Full AC, a winding-zero pattern with nonzero integral classical axis self-value, and the actual finite-stage infection geometry and coefficient maps. Signature defects use the oriented zero-surgery and first-positive-push conventions.

[A1]

Full AC is the explicit convention in the integral half-solution Definition and in the analytic suppliers. The Axiom of Choice, Integral half-solutions, Strebel classes, and L²-signatures

[F1]

For every fixed smooth zero surgery $M$ , the actual smooth bounding-defect invariant has a finite group-independent bound; its value is independent of smooth bounds and injective target enlargements, including arbitrary target groups and nononto representations. Smooth group signature defects are independent and uniformly bounded

[F2]

For any finite distinct odd primes and positive heights, explicit finite disk-band knots have nonnegative signature functions whose own selected primitive pair has value $2 N_{i}$ exceeding its height, whose other selected roots have value zero, and whose other selected-prime means vanish. Their actual canonical finite-cyclic defects are $D_{i} = 4 N_{i} / p_{i}$ . Finite prime families admit explicit separating knot signature bumps, Canonical finite cyclic knot signature defects equal Seifert signature averages

[F3]

The complete smooth infection assembly supplies actual full-forest quotient maps, the zero signature defects of all the actual joining/infection pieces, subgroup induction on the same canonical finite-cyclic bounds and the actual multi-boundary smooth rho identity. In the three-branch construction its distinguished negative branch has trivial restriction, and one or two positive branches have exact order $p$ . Finite-prime smooth infection obstructions use actual coefficients

[F4]

For the locally flat TOP application, the accepted finite-projective signature/gluing and actual lifted-half-pair vanishing apply to the supplied TOP bound. The boundary defect is identified with its smooth bounding-defect value by Bordism independence of the L²-signature defect, whose closed TOP equality is the explicitly external Lück–Schick input. Thus the same smooth uniform bound applies to that boundary invariant; no smoothing of the given four-manifold is used. Topological finite-projective L2 duality and whole-component boundary gluing, Topological lifted half-pairs force amenable signature defect zero, Cheeger–Gromov uniform bound for rho invariants

[F5]

On a supplied TOP initial integral-pair bordism, the full finite-forest survival theorem provides the same actual quotient and bottom meridian orders; on a supplied TOP homology-circle bound, the three-branch theorem provides the quotient killing the distinguished negative branch and leaving at least one positive branch nontrivial. These are supplied-data topological statements, independent of smooth bounding invariance. Integral-pair persistence and Blanchfield bordism for iterated infection, Unimodular infection-tower data force mixed-derived meridian survival, Three-branch infection data admit the nonhomomorphism coefficient quotients

Proof

Given: The actual finite-stage data and AC. First take the smooth bounding-defect application.

1.1A1F1F2F3givenconstructalgebra

Fix the stage $n \geq 1$ and desired family size $m$ . Choose $L > 0$ greater than [F1]'s bound for the one fixed pattern surgery $M (P (U))$ . The integral axis reduction has one finite exceptional-prime set, already checked on the actual coefficient maps in [F3]. Choose distinct odd $p_{1}, \dots, p_{m}$ outside that set, and apply [F2] at heights $p_{i} m n L / 2$ . Its actual first-push surfaces give $2 N_{i} > p_{i} m n L / 2$ , hence $D_{i} = 4 N_{i} / p_{i} > m n L$ , with zero selected cross-prime means. This proves the normalized inequality for the actual companion term; a large point-signature value alone, without its factor of $p_{i}$ , would not prove it.

2.1F1F2F3step 1.1constructalgebra

For any nonzero signed coefficient vector $(a_{i})$ , choose $i_{*}$ with maximal $∣ a_{i} ∣$ . The simultaneous finite-forest map of [F3] gives order exactly $p_{i_{*}}$ on every actual bottom meridian. Its restriction there is cyclic, and its extension over the canonical companion bound has that same positive meridian image. Actual same-complex subgroup induction and smooth independence in [F3] therefore make the signed bottom sum exactly $a_{i_{*}} D_{i_{*}}$ ; changing a cyclic generator only permutes the normalized roots. There are exactly $n \sum_{i} ∣ a_{i} ∣$ remaining signed pattern terms, each of absolute value below $L$ . Their sum has absolute value below $n L \sum_{i} ∣ a_{i} ∣ \leq n m L ∣ a_{i_{*}} ∣ < ∣ a_{i_{*}} ∣ D_{i_{*}}$ . The selected bottom sum thus has the fixed sign of $a_{i_{*}}$ and cannot be cancelled. This is the precise domination used in Theorem A.

3.1F1F2F3step 1.1step 2.1constructalgebra

For Theorem B choose one allowed odd prime and height $p (3 n L) / 2$ , so $D = 4 N / p > 3 n L$ . The actual three-branch map in [F3] kills the distinguished negative $J # J$ term. Each of its two positive bottom branches has value zero or $D$ , and at least one is $D$ . Their sum is $s D$ , $s \in {1, 2}$ , whereas the $3 n$ remaining pattern terms have total absolute value below $3 n L < D$ . The sum is therefore strictly positive. All asserted choices are finite and no coprime Alexander assumption has entered either construction. The separate source statement in section6 compares two different patterns under relative primeness of their closure Alexander polynomials; that hypothesis was neither used nor silently transferred here.

4.1F1F2F3F4F5step 1.1step 2.1step 3.1algebra∎

For a supplied TOP initial bound, [F5] supplies the actual quotient maps and bottom orders, and [F4] gives its actual half-pair zero defect and identifies every remaining boundary term with the invariant bounded in [F1]. The standard joining/infection pieces remain the supplied smooth pieces; their coefficients and canonical cyclic comparisons in [F3] are unchanged. Hence the exact same finite signed boundary identity and the inequalities in steps1.1–3.1 give the fixed sign and domination for the TOP application. The only newly imported assertion is the closed TOP equality explicitly recorded in [F4], not the analytic uniform estimate or the coefficient survival calculation. This completes every stated finite-stage selection and separation conclusion.

Depends on

Used by

Infinite rank for iterated winding-zero satellites Theorem

Dependency tree · two levels

58 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Jae Choon Cha and Taehee Kim, “Iterated satellite operators on the knot concordance group” (standard reference, not scraped)
Jae Choon Cha and Charles Livingston, “Knot signature functions are independent” (standard reference, not scraped)
Jae Choon Cha, “Structure of the string link concordance group and Hirzebruch-type invariants” (standard reference, not scraped)