# Difference between revisions of "ApCoCoA-1:Knot group"

### Figure Eight Group

#### Description

Knots are in mathematic embedding of the circleline in a three-dimensional sphere. The figure eight knot has the following presentation:

#### Reference

Michael Eisermann, Knotengruppen-Darstellungen und Invarianten von endlichem Typ, Rheinischen Friedrich-Wilhelms-Universität, Bonn, 2000

#### Computation

``` /*Use the ApCoCoA package ncpoly.*/

Use ZZ/(2)[a,b,c,d];
NC.SetOrdering("LLEX");

Define CreateRelationsAchterknoten()
Relations:=[];
Append(Relations,[[a,c],]);
Append(Relations,[[c,a],]);
Append(Relations,[[b,d],]);
Append(Relations,[[d,b],]);

// add the relation a^(-1)bab^(-1)ab = ba^(-1)ba
Append(Relations,[[c,b,a,d,a,b],[b,c,b,a]]);
Return Relations;
EndDefine;

Relations:=CreateRelationsAchterknoten();
Relations;

Gb:=NC.GB(Relations,31,1,100,1000);
Gb;

```

#### Examples in Symbolic Data Format

``` <FREEALGEBRA createdAt="2014-07-03" createdBy="strohmeier">
<vars>a,b,c,d</vars>
<uptoDeg>12</uptoDeg>
<basis>
<ncpoly>a*c-1</ncpoly>
<ncpoly>c*a-1</ncpoly>
<ncpoly>b*d-1</ncpoly>
<ncpoly>d*b-1</ncpoly>
<ncpoly>c*b*a*d*a*b-b*c*b*a</ncpoly>
</basis>
<Comment>The partial LLex Gb has 316 elements</Comment>
<Comment>Achterknotengruppe</Comment>
</FREEALGEBRA>
```