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

Clover leaf loop

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 . ${\ displaystyle K \ subset S ^ {3}}$ ${\ displaystyle \ mathbb {R} ^ {3}}$ ${\ displaystyle (\ mathbb {R} ^ {3}) ^ {*} = S ^ {3}}$ ${\ displaystyle K}$ ${\ displaystyle S ^ {3} \ setminus K}$

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 ${\ displaystyle S ^ {3} \ setminus K}$ ${\ displaystyle N (K)}$ ${\ displaystyle K}$

{\ displaystyle X_ {K} = S ^ {3} \ setminus int (N (K))}

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

{\ displaystyle H_ {0} (X_ {K}) = \ mathbb {Z}, \; H_ {1} (X_ {K}) = \ mathbb {Z}, \; H_ {2} (X_ {K} ) = 0, \; H_ {3} (X_ {K}) = 0}

.

(Because and are homotopy equivalent , these invariants do not depend on which of the two definitions above is used.) ${\ displaystyle X_ {K}}$ ${\ displaystyle S ^ {3} \ setminus K}$

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. ${\ displaystyle S ^ {3} \ setminus K_ {1}}$ ${\ displaystyle S ^ {3} \ setminus K_ {2}}$ ${\ displaystyle K_ {1}}$ ${\ displaystyle K_ {2}}$

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 .

Individual evidence

^ Wilbur Whitten: Knot complements and groups. Topology, 26, nos. 1, 41-44 (1987).
↑ C. McA. Gordon, J. Luecke: Knots are determined by their complements. J. Amer. Math. Soc. 2 (1989) no. 2, 371-415.

[1] Wilbur Whitten: Knot complements and groups. Topology, 26, nos. 1, 41-44 (1987).

[2] C. McA. Gordon, J. Luecke: Knots are determined by their complements. J. Amer. Math. Soc. 2 (1989) no. 2, 371-415.