Difference between revisions of "ApCoCoA-1:Torus Knot Group"
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With r(phi) = ( (cos(q*phi)+2)*sin(p*phi) ) 0<=phi<=2pi | With r(phi) = ( (cos(q*phi)+2)*sin(p*phi) ) 0<=phi<=2pi | ||
sin(q*phi) | 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 > | tng(a,b)= < a,b| a^p = b^q = 1 > | ||
==== Reference ==== | ==== Reference ==== | ||
Michael Eisermann, Knotengruppen-Darstellungen und | Michael Eisermann, Knotengruppen-Darstellungen und | ||
| − | Invarianten von endlichem Typ, Rheinischen Friedrich-Wilhelms-Universität, Bonn, 2000; Johannes Diernke, Universität Oldenburg, Oldenburg, 2010/2011: [http://www.informatik.uni-oldenburg.de/~trigger/content/opengl/opengl_course/slides/2010-JOGL-08-Torus-Knoten.pdf] | + | Invarianten von endlichem Typ, Rheinischen Friedrich-Wilhelms-Universität, Bonn, 2000; Johannes Diernke, Universität Oldenburg, Oldenburg, 2010/2011: [http://www.informatik.uni-oldenburg.de/~trigger/content/opengl/opengl_course/slides/2010-JOGL-08-Torus-Knoten.pdf]; [http://mathworld.wolfram.com/TorusKnot.html] |
==== Computation ==== | ==== Computation ==== | ||
| Line 17: | Line 17: | ||
// Define the variable q,p of the Torusknotengroup | // Define the variable q,p of the Torusknotengroup | ||
| + | //ggT(q,p)=1 | ||
MEMORY.P := 2; | MEMORY.P := 2; | ||
MEMORY.Q := 3; | MEMORY.Q := 3; | ||
| − | + | ||
Use ZZ/(2)[a,b,c,d]; | Use ZZ/(2)[a,b,c,d]; | ||
NC.SetOrdering("LLEX"); | NC.SetOrdering("LLEX"); | ||
| Line 42: | Line 43: | ||
Gb:=NC.GB(Relations,31,1,100,1000); | Gb:=NC.GB(Relations,31,1,100,1000); | ||
Gb; | Gb; | ||
| + | |||
====Examples in Symbolic Data Format==== | ====Examples in Symbolic Data Format==== | ||
<FREEALGEBRA createdAt="2014-07-03" createdBy="strohmeier"> | <FREEALGEBRA createdAt="2014-07-03" createdBy="strohmeier"> | ||
Latest revision as of 08:27, 20 July 2014
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>
