Node group

In knot theory , a branch of mathematics , a circle embedded in Euclidean space is called a knot . The corresponding node group is then the fundamental group of the complement of the node ${\displaystyle K}$

${\displaystyle \pi _{1}(\mathbb {R} ^{3}\setminus K).}$

Different convention

In topology , instead of Euclidean space, one often considers its one-point compactification and, accordingly, nodes as embedded circles in the . ${\displaystyle \mathbb {S} ^{3}}$ ${\displaystyle \mathbb {S} ^{3}}$

It can be shown that the resulting nodal group

${\displaystyle \pi _{1}(\mathbb {S} ^{3}\setminus K)}$

is isomorphic to . ${\displaystyle \pi _{1}(\mathbb {R} ^{3}\setminus K)}$

properties

Equivalent nodes have isomorphic node groups, so the node group is a node invariant and can be used to distinguish nodes.

However, the converse is not true, so there are non-equivalent nodes with isomorphic node groups. It is also an algorithmically difficult problem to prove the non-isomorphism of nodal groups.

The abelization of the node group is always isomorphic to the group of whole numbers . This follows from Alexander's duality principle . ${\displaystyle \mathbb {Z} }$

The node group can be calculated quite easily with the Wirtinger algorithm . (That is, the Wirtinger algorithm provides a finite presentation of the node group.) However, there is no general algorithm that decides for two finite group presentations whether the groups are isomorphic.

All generators in the Wirtinger presentation are meridians of the knot and, in particular, all these generators are conjugated to one another. Under the abelization map, all are mapped to the same generator of the integers . ${\displaystyle \mathbb {Z} }$

Examples

• The node group of the trivial node is . ${\displaystyle \mathbb {Z} }$
• The knot group of the trefoil knot is the braid group with presentation ${\displaystyle B_{3}}$

${\displaystyle \langle x,y\mid x^{2}=y^{3}\rangle }$or . ${\displaystyle \langle a,b\mid aba=bab\rangle }$

• The knot group of the (p, q) - torus knot is

${\displaystyle \langle x,y\mid x^{p}=y^{q}\rangle }$.

• The node group of the figure eight is

${\displaystyle \langle x,y\mid x^{-1}yxy^{-1}xy=yx^{-1}yx\rangle }$.

literature

• Burde, Gerhard; Zieschang, Heiner. Knots. Second edition. de Gruyter Studies in Mathematics, 5. Walter de Gruyter & Co., Berlin, 2003. xii + 559 pp. ISBN 3-11-017005-1

Web links

• Hazewinkel, Michiel, ed. (2001), " Knot and Link Groups ", Encyclopedia of Mathematics , Springer, ISBN 978-1556080104