Knot group

In knot theory, a branch of mathematics, one calls an embedded circle in the Euclidean space as a node. The corresponding node group is then the fundamental group of the complement of the node

Other Convention

In topology one considers instead of the Euclidean space is often its one-point compactification and corresponding node as embedded circles in the.

It can be shown that the so resulting node set

Is isomorphic to.

Properties

Equivalent nodes have isomorphic knot groups, the node groups is therefore a knot invariant and can serve to distinguish nodes.

However, the converse is not true, as there are non- equivalent nodes with isomorphic knot groups. It is also an algorithmically difficult problem of proving the non- isomorphism of node groups.

The Abelianisierung the node group is always isomorphic to the group of integers. This follows from the Alexander 's duality theorem.

The node group can be calculated quite simply with a Wirtinger algorithm. ( That is, the Wirtinger algorithm provides a finite presentation of the group of nodes. ) But there is no general algorithm which decides to two finite group presentations, whether the groups are isomorphic.

Examples

• The node set of trivial node.
• The node group of the trefoil knot, the braid group with presentation

• The node set of (p, q) - Torusknotens

• The node set of the eight node