User:Shawkm/06-1350-HW4: Difference between revisions

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===Here Goes Something New(?)===
===Here Goes Something New(?)===


The first thing to notice is that the relation <math>\rho_3 </math> holds for <math> b^+ <\math> and <math> b^- <\math> so we have another version:
The first thing to notice is that the relation <math>\rho_3 </math> holds for <math> b^+ </math> and <math> b^- </math> so we have another version:

{| align=center
{| align=center
|-
|-
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|}
|}


This is probably a completely trivial remark, and it is also trivial to see that mathematica will deal with this in the same way as for <math> b^+ <\math> so I won't verify that <math> d^2 =0 <\math>.</math>
This is probably a completely trivial remark, and it is also trivial to see that mathematica will deal with this in the same way as for <math> b^+ </math> so I won't verify that <math> d^2 =0 </math>.


What I will do now is linearize and write down in functional form the relation R2. We know R2 correspnds to <math>(123)^\star B^\pm\cdot(132)^\star B^\mp=1_3</math>. Rewriting it gives,

{| align=center
|-
|<math>\rho_2(x_1, x_2, x_3) = </math>
|<math>b^+(x_1,x_2,x_3) + b^-(x_1,x_3,x_2)</math>
|}

or

{| align=center
|-
|<math>\rho_2(x_1, x_2, x_3) = </math>
|<math>b^-(x_1,x_2,x_3) + b^+(x_1,x_3,x_2)</math>
|}

Latest revision as of 19:38, 2 December 2006

The Generators

Our generators are , , and :

Picture 06-1350-BPlus.svg
Generator
Perturbation

The Relations

The Reidemeister Move R3

The picture (with three sides of the shielding removed) is

06-1350-R4.svg

In formulas, this is

.

Linearized and written in functional form, this becomes

The Syzygies

The "B around B" Syzygy

The picture, with all shielding removed, is

06-1350-BAroundB.svg
(Drawn with Inkscape)
(note that lower quality pictures are also acceptable)

The functional form of this syzygy is

A Mathematica Verification

The following simulated Mathematica session proves that for our single relation and single syzygy, . Copy paste it into a live Mathematica session to see that it's right!

In[1]:= d1 = { rho3[x1_, x2_, x3_, x4_] :> bp[x1, x2, x3] + bp[x1 + x3, x2, x4] + bp[x1, x3, x4] - bp[x1 + x2, x3, x4] - bp[x1, x2, x4] - bp[x1 + x4, x2, x3] }; d2 = { BAroundB[x1_, x2_, x3_, x4_, x5_] :> rho3[x1, x2, x3, x5] + rho3[x1 + x5, x2, x3, x4] - rho3[x1 + x2, x3, x4, x5] - rho3[x1, x2, x4, x5] - rho3[x1 + x4, x2, x3, x5] - rho3[x1, x2, x3, x4] + rho3[x1, x3, x4, x5] + rho3[x1 + x3, x2, x4, x5] };
In[3]:= BAroundB[x1, x2, x3, x4, x5] /. d2
Out[3]= - rho3[x1, x2, x3, x4] + rho3[x1, x2, x3, x5] - rho3[x1, x2, x4, x5] + rho3[x1, x3, x4, x5] - rho3[x1 + x2, x3, x4, x5] + rho3[x1 + x3, x2, x4, x5] - rho3[x1 + x4, x2, x3, x5] + rho3[x1 + x5, x2, x3, x4]
In[4]:= BAroundB[x1, x2, x3, x4, x5] /. d2 /. d1
Out[4]= 0


Here Goes Something New(?)

The first thing to notice is that the relation holds for and so we have another version:

This is probably a completely trivial remark, and it is also trivial to see that mathematica will deal with this in the same way as for so I won't verify that .


What I will do now is linearize and write down in functional form the relation R2. We know R2 correspnds to . Rewriting it gives,

or

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