ApCoCoA-1:Torus Knot Group

Torus Knot Group

Description

The Torus Knot Group is described by a space curve r(phi).

                (cos(q*phi)+2)*cos(p*phi) 
With r(phi) = ( (cos(q*phi)+2)*sin(p*phi) ) 0<=phi<=2pi
                        sin(q*phi)

By interlacing a string through the hole of a torus p times with q revolutions, a (p,q)-torus knot results. The ends of the string are joined together. The variables p,q are relatively prime. And has the following presentation:

tng(a,b)= < a,b| a^p = b^q = 1 >

Reference

Michael Eisermann, Knotengruppen-Darstellungen und Invarianten von endlichem Typ, Rheinischen Friedrich-Wilhelms-Universität, Bonn, 2000; Johannes Diernke, Universität Oldenburg, Oldenburg, 2010/2011: [1]; [2]

Computation

/*Use the ApCoCoA package ncpoly.*/

// Define the variable q,p of the Torusknotengroup
//ggT(q,p)=1
MEMORY.P := 2;
MEMORY.Q := 3;
 
Use ZZ/(2)[a,b,c,d];
NC.SetOrdering("LLEX");

Define CreateRelationsTorusknoten()
  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^p = b^q 
  Append(Relations,[[a^MEMORY.P],[b^MEMORY.Q]]);
   Return Relations;
EndDefine;

Relations:=CreateRelationsTorusknoten();
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>14</uptoDeg>
 	<basis>
 	<ncpoly>a*c-1</ncpoly>
 	<ncpoly>c*a-1</ncpoly>
 	<ncpoly>b*d-1</ncpoly>
 	<ncpoly>d*b-1</ncpoly>
 	<Comment>Relation: a^p=b^q</Comment>
 	<ncpoly>a*a-b*b*b</ncpoly> 
 	</basis>
 	<Comment>The partial LLex Gb has 198 elements</Comment>
 	<Comment>Torusknotengruppe_p2q3</Comment>
 	<Comment>Torusknotengruppe_p2q3 is isomorph to "Trefoil Knot Group"</Comment>
 </FREEALGEBRA>