DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedverified 2026-10-22 (gpt-6.1-sol)

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SL2(C)-abelian surgery

Definition

Assume full AC. Let $K \subset S^{3}$ be a smooth oriented knot, let $p, q \in Z$ be coprime and not both zero, and let $E_{K} (p / q)$ be the Dehn filling whose attached solid-torus meridian is identified with $p [μ] + q [λ]$ . A closed 3-manifold $Y$ is $S L_{2} (C)$ -abelian if every homomorphism $π_{1} (Y) \to S L_{2} (C)$ has abelian image. The slope $p / q$ is an $S L_{2} (C)$ -abelian surgery slope for $K$ if $E_{K} (p / q)$ is $S L_{2} (C)$ -abelian. The knot admits infinitely many such surgeries if infinitely many distinct rational slopes have this property.

Definition references

The primitive slope and filling convention

Baldwin and Sivek, §§1 and 2.2

Sivek and Zentner, §1

Depends on

Used by

Dependency tree · two levels

30 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

John A. Baldwin and Steven Sivek, Torus knots, the A-polynomial, and SL(2,C), §§1 and 2.2 (standard reference, not scraped)
Steven Sivek and Raphael Zentner, SU(2)-cyclic surgeries and the pillowcase, §1 (standard reference, not scraped)