LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)

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Noncyclic SU(2) representations of numerator-one surgeries

Statement

Assume the Axiom of Choice. Let $K \subset S^{3}$ be a nontrivial smooth knot, and let $p, q$ be coprime integers with $∣ p ∣ = 1$ and $q \neq 0$ . For the Dehn filling $Y_{p / q} = E_{K} (p / q)$ whose attached solid-torus meridian is identified with $p [μ] + q [λ]$ , the fundamental group $π_{1} (Y_{p / q})$ admits a homomorphism to $S U (2)$ with noncyclic image.

Statement references

Chosen knot exterior

Meridian and preferred longitude

Filling slope convention

Ordinary closed instanton surgery data

Kronheimer–Mrowka, Dehn surgery, the fundamental group and SU(2), Theorem 1 and §§2–4

Kronheimer–Mrowka, Witten's conjecture and Property P, §§1 and 3

Kronheimer–Mrowka, Knots, sutures and excision, §§7.5 and 7.8

Motegi, Dehn surgery on knots: survey, pp. 19–21 and 26

Kronheimer–Mrowka, published Theorem 1, p. 741

Facts & Assumptions

Given: Full AC, a nontrivial smooth knot $K \subset S^{3}$ , and coprime integers $p, q$ with $∣ p ∣ = 1$ and $q \neq 0$ .

[F1]

External input. For a nontrivial knot $K \subset S^{3}$ and rational surgery coefficient $r$ with $∣ r ∣ \leq 2$ , Kronheimer–Mrowka's published Theorem 1 states that the surgery group admits a homomorphism to $S U (2)$ with noncyclic image (and consequently is not cyclic). Source: Dehn surgery, the fundamental group and SU(2), Mathematical Research Letters 11 (2004), 741–754, Theorem 1, p. 741, DOI 10.4310/mrl.2004.v11.n6.a3. This is an imported theorem, not a locally reconstructed gauge-theoretic proof.

[F2]

Rational coefficient $p / q$ denotes filling with meridian $p [μ] + q [λ]$ in the meridian and preferred-longitude convention; simultaneous sign change leaves the unoriented slope unchanged. Oriented slopes and knot Dehn filling Meridian and preferred longitude

[A1]

Full AC is retained as assumed in the Statement. The Axiom of Choice

Proof

1.1givenF2algebra

Since $q$ is a nonzero integer, $∣ q ∣ \geq 1$ . Thus $r = p / q$ is rational and satisfies $∣ r ∣ = 1 / ∣ q ∣ \leq 1 \leq 2$ . By [F2], the manifold in the Statement is precisely the coefficient- $r$ surgery appearing in [F1]. Both signs of $r$ are covered by its absolute-value hypothesis.

2.1givenA1F1step 1.1∎

Apply [F1] to this nontrivial knot and this coefficient. It gives a homomorphism $π_{1} (Y_{p / q}) \to S U (2)$ with noncyclic image, as required.

External input and reading limits

The published primary PDF was obtained from the International Press journal archive. Theorem 1 on printed p. 741 (PDF p. 1) was read directly, including its exact nontrivial-knot and $∣ r ∣ \leq 2$ hypotheses and its noncyclic-image conclusion. The slope definitions and $[p a + q b]$ filling convention on pp. 743 and 745–746 were inspected, together with the application and nonvanishing discussion on pp. 752–753. These source readings identify the imported theorem and its convention; they do not constitute a local proof audit of its gauge theory or of all its cited prerequisites.

The paper explicitly says on p. 741 that its proof avoids explicit instanton Floer homology and the Floer exact triangle, using holonomy perturbations instead. Accordingly, this item's historical identifier does not assert a general surgery-triangle or rank theorem: its full Statement is the numerator-one representation conclusion proved above from the external input. The retained singular-instanton complex and integral resolution cube keep their existing scope; neither is relabelled as supplying this closed-surgery result. Ordinary KHI nonvanishing and varying-meridian representation paths are separate inputs and are not inferred here.

Depends on

Used by

Property P for nontrivial knots Theorem

Dependency tree · two levels

23 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

P. B. Kronheimer and T. S. Mrowka, Dehn surgery, the fundamental group and SU(2), Mathematical Research Letters 11 (2004), 741–754, Theorem 1, p. 741 (standard reference, not scraped)
P. B. Kronheimer and T. S. Mrowka, Dehn surgery, the fundamental group and SU(2), Theorem 1 and §§2–4 (standard reference, not scraped)
P. B. Kronheimer and T. S. Mrowka, Witten's conjecture and Property P, §§1 and 3 (standard reference, not scraped)
P. B. Kronheimer and T. S. Mrowka, Knots, sutures and excision, §§7.5 and 7.8 (standard reference, not scraped)
Kimihiko Motegi, Dehn surgery on knots: survey, pp. 19–21 and 26 (standard reference, not scraped)