Node group

A node group is a characteristic of a node, defined as the fundamental group of its add-on.

Content

Definition

Let be $K\subset \mathbb {R} ^{3}$ ${\ displaystyle K \ subset \ mathbb {R} ^ {3}}$ there is a knot. Then the node node group is defined as the fundamental group. $\pi _{1}(\mathbb {R} ^{3}\setminus K)$ ${\ displaystyle \ pi _ {1} (\ mathbb {R} ^ {3} \ setminus K)}$ . ^[1] .

Comment

According to other agreements, a node is considered to be an embedding of a circle in a 3-sphere . In this case, the node group is defined as the fundamental group of its complement in $S^{3}$ ${\ displaystyle S ^ {3}}$ . Both definitions give isomorphic groups.

Properties

Two equivalent knots have isomorphic knot groups, so a knot group is an invariant of a knot and can be used to establish the nonequivalence of a pair of knots. However, two nonequivalent nodes may have isomorphic groups of nodes (see the example below).

Abelization of a node group is always isomorphic to an infinite cyclic group. $\mathbb {Z}$ ${\ displaystyle \ mathbb {Z}}$ . This follows from the fact that abelization coincides with the first homology group , which is easy to calculate.

The group of nodes (as well as the fundamental group of oriented links in the general case) can be calculated using relatively simple algorithms using .

Examples

The trivial knot group is isomorphic $Z$ {\ displaystyle \ mathbb {Z}} .
- The reverse is also true.
The trefoil group is isomorphic to the braid group $B^{3}$ ${\ displaystyle B ^ {3}}$ This group has a task :
$\langle x,y\mid x^{2}=y^{3}\rangle$ ${\ displaystyle \ langle x, y \ mid x ^ {2} = y ^ {3} \ rangle}$ or $\langle a,b\mid aba=bab\rangle$ ${\ displaystyle \ langle a, b \ mid aba = bab \ rangle}$ .
Group $(p,q)$ ${\ displaystyle (p, q)}$ - the torus knot possesses the task:
$\langle x,y\mid x^{p}=y^{q}\rangle$ ${\ displaystyle \ langle x, y \ mid x ^ {p} = y ^ {q} \ rangle}$ .
The group of eight has the task:
$\langle x,y\mid yxy^{-1}xy=xyx^{-1}yx\rangle$ ${\ displaystyle \ langle x, y \ mid yxy ^ {- 1} xy = xyx ^ {- 1} yx \ rangle}$ .
The direct knot and the babi knot have isomorphic groups of knots, but these knots are not equivalent.

Notes

↑ Boltyansky, 1982 , p. 119.

Literature

Group knots and links - an article from the Mathematical Encyclopedia
Boltyansky V.G. , Efremovich V.A. Visual topology. - M .: Science, 1982. - 160 p.

[_c85dd322f0547739-1] Boltyansky, 1982 , p. 119.