Dehn Surgery: Constructing 3-Manifolds
Cut out a knot's tubular neighborhood, reglue with slope p/q — Lickorish-Wallace: every closed 3-manifold arises this way
Select a knot and surgery coefficient to see the resulting 3-manifold
Dehn surgery: Remove a solid torus neighborhood of knot K ⊂ S³. The boundary torus has basis {meridian μ, longitude λ}. Reglue a solid torus mapping its meridian to pμ + qλ.
Lickorish-Wallace (1960): Every closed orientable 3-manifold is surgery on a link in S³.
Kirby calculus: Two surgery diagrams give the same 3-manifold iff related by Kirby moves (handle slides, stabilizations).
Key results: (+1)-surgery on trefoil → Poincaré homology sphere (π₁ has order 120). 0-surgery → S¹×S². p/q surgery on unknot → lens space L(p,q).