Dehn surgery

In topology, a branch of mathematics, Dehn surgery is a returning to Max Dehn method of constructing 3-dimensional manifolds.

Definition

Let be a 3-manifold and an embedding with image. Be an integer matrix. They stick on to that by identifying.

One can show that the so constructed manifold ( not ) depends up to homeomorphism only by the node and the numbers. This is known as the manifold obtained by Dehn surgery on knots with coefficients.

Correspondingly, one can define coefficients at the nodes for entanglement a manifold by sequential execution (in any order ) of the Dehn surgeries.

Construction of 3-manifolds ( set of Lickorish - Wallace )

Every closed, orientable, connected 3-manifold can be constructed by Dehn surgery on a link in the 3- sphere. It is even possible to ensure that all components of all coefficients and unknotted.

Construction of hyperbolic 3-manifolds (Theorem of Thurston )

When a complete hyperbolic metric of finite volume contributes, then, almost all by Dehn surgery on manifolds generated also hyperbolic.

For the figure-eight knot, there are 10 exceptional ( ie: non- hyperbolic ) Dehn surgeries. Lackenby and Meyerhoff have proved that the number of exceptional Dehn surgeries is for each node a maximum of 10.

Documents

• Geometric topology
• Knot theory