Node complement
In knot theory , a branch of mathematics , the knot complement is the space that remains after removing a knot from the 3-sphere .
definition
Let it be a knot . (The case of knots in Euclidean space can be reduced to the first case by considering the one-point compactification .) The complement of the knot is then the open manifold .
Instead of the open manifold, one often considers the manifold with a boundary constructed as follows . Let be a tube environment of , i.e. homeomorphic to the full torus . Then
a compact 3-manifold whose boundary is a torus .
The complement of links can be defined analogously .
Invariants
The fundamental group of Knotenkomplements is the node set , it can with the Wirtinger algorithm presents are. For prime nodes , the node complement is uniquely determined by the node group.
The homology groups of the node complement do not depend on the node, it applies
- .
(Because and are homotopy equivalent , these invariants do not depend on which of the two definitions above is used.)
Gordon-Luecke's theorem
A theorem proved by Gordon and Luecke says that nodes are uniquely determined by their complement: If is homeomorphic to , then the nodes and are isotopic . The corresponding statement for entanglements does not apply.
literature
- Heinrich Tietze: About the topological invariants of multidimensional manifolds. Monthly Math. Phys. 19 (1908), no. 1, 1-118.
- C. McA. Gordon, J. Luecke: Knots are determined by their complements. Bull. Amer. Math. Soc. (NS) 20 (1989) no. 1, 83-87.
- Colin Adams: The Knot Book. Spektrum Akademischer Verlag, Heidelberg 1995, ISBN 3-86025-338-7 .