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===The Generators===
===The Generators===


Our generators are <math>T</math>, <math>R</math>, <math>Y=\Phi</math> and <math>B^{\pm}</math>. (You might consider splitting <math>Y</math> into two, <math>Y^{up}</math> and <math>Y^{dn}</math>). In pictures, they are
Our generators are <math>T</math>, <math>R</math>, <math>Y=\Phi</math> and <math>B^{\pm}</math> (we might consider splitting <math>Y</math> into two, <math>Y^{up}</math> and <math>Y^{dn}</math>):
{| align=center
{| align=center
|- align=center
|- align=center valign=middle
|align=left|Picture
|
|
|
|
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|[[Image:06-1350-BPlus.svg|100px]]
|[[Image:06-1350-BPlus.svg|100px]]
|
|
|- align=center
|- align=center valign=middle
|align=left|Generator
|<math>T</math>
|<math>T</math>
|<math>R</math>
|<math>R</math>
Line 17: Line 19:
|<math>B^+</math>
|<math>B^+</math>
|<math>B^-</math>
|<math>B^-</math>
|- align=center valign=middle
|align=left|Perturbation
|<math>t</math>
|<math>r</math>
|<math>y^u</math>
|<math>y^d</math>
|<math>b^+</math>
|<math>b^-</math>
|}
|}


Revision as of 15:27, 18 November 2006

The Generators

Our generators are [math]\displaystyle{ T }[/math], [math]\displaystyle{ R }[/math], [math]\displaystyle{ Y=\Phi }[/math] and [math]\displaystyle{ B^{\pm} }[/math] (we might consider splitting [math]\displaystyle{ Y }[/math] into two, [math]\displaystyle{ Y^{up} }[/math] and [math]\displaystyle{ Y^{dn} }[/math]):

Picture 06-1350-BPlus.svg
Generator [math]\displaystyle{ T }[/math] [math]\displaystyle{ R }[/math] [math]\displaystyle{ Y^{up} }[/math] [math]\displaystyle{ Y^{dn} }[/math] [math]\displaystyle{ B^+ }[/math] [math]\displaystyle{ B^- }[/math]
Perturbation [math]\displaystyle{ t }[/math] [math]\displaystyle{ r }[/math] [math]\displaystyle{ y^u }[/math] [math]\displaystyle{ y^d }[/math] [math]\displaystyle{ b^+ }[/math] [math]\displaystyle{ b^- }[/math]

The Relations

The Reidemeister Move R3

The picture is

06-1350-R4.svg

In formulas, this is

[math]\displaystyle{ (1230)^\star B^+ (1213)^\star B^+ (1023)^\star B^+ = (1123)^\star B^+ (1203)^\star B^+ (1231)^\star B^+ }[/math].

Thus the R3 component of [math]\displaystyle{ d }[/math] is

[math]\displaystyle{ (1230)^\star b^+ +(1213)^\star b^+ +(1023)^\star b^+ -(1123)^\star b^+ -(1203)^\star b^+ -(1231)^\star b^+ }[/math].

The Syzygies

A Mathematica Verification

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