ApCoCoA-1:Knot Group

Figure Eight Group

Description

In mathematics knots are embeddings of the circleline in a three-dimensional sphere. And the figure eight knot is a specific knot with crossingnumber four. And has the following presentation:

feg(x,y)= < x,y | yxy^{-1}xy=xyx^{-1}yx >

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:=[];
    //add the inverse relations
    Append(Relations,[[a,c],[1]]);
    Append(Relations,[[c,a],[1]]);
    Append(Relations,[[b,d],[1]]);
    Append(Relations,[[d,b],[1]]);
    
    // 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>