Dehn Surgery on Knots

TLDR

Dehn surgery constructs 3‑manifolds by removing a solid torus around a knot in \(S^3\) and re‑gluing it via a chosen slope \(r\) on the boundary torus. The paper aims to deepen understanding of Dehn surgery to advance knowledge of 3‑manifold structure, extending the technique from knots to arbitrary manifolds with torus boundary. The authors define \(r\)-Dehn surgery on a knot or link as gluing a solid torus along a slope \(r\) on the boundary torus, producing a closed oriented 3‑manifold, and generalize this to \(r\)-Dehn filling on any compact irreducible 3‑manifold with an incompressible torus boundary. Dehn surgery on the trefoil produces infinitely many non‑simply‑connected homology spheres, and in fact every closed oriented 3‑manifold can be obtained by De.

Abstract

In [D], Dehn considered the following method for constructing 3-manifolds: remove a solid torus neighborhood N(K) of some knot X in the 3-sphere S and sew it back differently. In particular, he showed that, taking X to be the trefoil, one could obtain infinitely many non-simply-connected homology spheres o in this way. Let Mx = S —N(K). Then the different resewings are parametrized by the isotopy class r of the simple closed curve on the torus oMK that bounds a meridional disk in the re-attached solid torus. We denote the resulting closed oriented 3-manifold by MK(), and say that it is obtained by r-Dehn surgery on X. More generally, one can consider the manifolds ML(*) obtained by r-Dehn surgery on a fc-component link L = Xi U • • • U Kk in S, where r = (r\,..., ;>). It turns out that every closed oriented 3-manifold can be constructed in this way [Wal, Lie]. Thus a good understanding of Dehn surgery might lead to progress on general questions about the structure of 3-manifolds. Starting with the case of knots, it is natural to extend the context a little and consider the manifolds M(r) obtained by attaching a solid torus V to an arbitrary compact, oriented, irreducible (every 2-sphere bounds a 3-ball) 3-manifold M with dM an incompressible torus, where r is the isotopy class (slope) on dM of the boundary of a meridional disk of V. We say that M(r) is the result of r-Dehn filling on M. An observed feature of this construction is that