Dehn surgery

In topology, a branch of mathematics, a Dehn surgery, named after Max Dehn, is a construction used to modify 3-manifolds. The process takes as input a 3-manifold together with a link. It is often conceptualized as two steps: drilling then filling (also known as Dehn-tistry).

We can pick two oriented simple closed curves m and ℓ on the boundary torus of the 3-manifold that generate the fundamental group of the torus. This gives any simple closed curve ${\displaystyle \gamma }$ on that torus two coordinates p and q, each coordinate corresponding to the algebraic intersection of the curve with m and ℓ respectively. These coordinates only depend on the homotopy class of ${\displaystyle \gamma }$.

We can specify a homeomorphism of the boundary of a solid torus to T by having the meridian curve of the solid torus map to a curve homotopic to ${\displaystyle \gamma }$. As long as the meridian maps to the surgery slope ${\displaystyle [\gamma ]}$, the resulting Dehn surgery will yield a 3-manifold that will not depend on the specific gluing (up to homeomorphism). The ratio p/q is called the surgery coefficient.

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