User:Drorbn/06-1350-HW4: Difference between revisions
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====The "B around B" Syzygy==== |
====The "B around B" Syzygy==== |
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The picture, with all shielding removed, is |
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|align=right|(Drawn with [http://www.inkscape.org/ Inkscape])<br>(note that lower quality pictures are also acceptable) |
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The functional form of this syzygy is |
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|<math>BB(x_1,x_2,x_3,x_4,x_5) = </math> |
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|<math>\rho_3(x_1, x_2, x_3, x_5) + \rho_3(x_1 + x_5, x_2, x_3, x_4) - \rho_3(x_1 + x_2, x_3, x_4, x_5)</math> |
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|<math>- \rho_3(x_1, x_2, x_4, x_5) - \rho_3(x_1 + x_4, x_2, x_3, x_5) - \rho_3(x_1, x_2, x_3, x_4)</math> |
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|<math>+ \rho_3(x_1, x_3, x_4, x_5) + \rho_3(x_1 + x_3, x_2, x_4, x_5).</math> |
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===A Mathematica Verification=== |
===A Mathematica Verification=== |
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The following simulated Mathematica session proves that for our single relation and single syzygy, <math>d^2=0</math>. Copy paste it into a live Mathematica session to see that it's right! |
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{{In|n=1|in=<nowiki>d1 = { |
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rho3[x1_, x2_, x3_, x4_] :> bp[x1, x2, x3] + bp[x1 + x3, x2, x4] + |
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bp[x1, x3, x4] - bp[x1 + x2, x3, x4] - bp[x1, x2, x4] - |
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bp[x1 + x4, x2, x3] |
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}; |
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d2 = { |
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BAroundB[x1_, x2_, x3_, x4_, x5_] :> rho3[x1, x2, x3, x5] + |
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rho3[x1 + x5, x2, x3, x4] - rho3[x1 + x2, x3, x4, x5] - |
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rho3[x1, x2, x4, x5] - rho3[x1 + x4, x2, x3, x5] - |
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rho3[x1, x2, x3, x4] + rho3[x1, x3, x4, x5] + |
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rho3[x1 + x3, x2, x4, x5] |
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};</nowiki>}} |
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{{InOut|n=3|in=<nowiki>BAroundB[x1, x2, x3, x4, x5] /. d2</nowiki>|out=<nowiki>- rho3[x1, x2, x3, x4] + rho3[x1, x2, x3, x5] - rho3[x1, x2, x4, x5] |
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+ rho3[x1, x3, x4, x5] - rho3[x1 + x2, x3, x4, x5] |
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+ rho3[x1 + x3, x2, x4, x5] - rho3[x1 + x4, x2, x3, x5] |
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+ rho3[x1 + x5, x2, x3, x4]</nowiki>}} |
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{{InOut|n=4|in=<nowiki>BAroundB[x1, x2, x3, x4, x5] /. d2 /. d1</nowiki>|out=<nowiki>0</nowiki>}} |
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Latest revision as of 18:41, 20 November 2006
The Generators
Our generators are , , and :
| Picture | 
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| Generator | |||||
| Perturbation |
The Relations
The Reidemeister Move R3
The picture (with three sides of the shielding removed) is
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
 |
| (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]:=
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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]
};
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In[3]:=
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BAroundB[x1, x2, x3, x4, x5] /. d2
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Out[3]=
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- 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]
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In[4]:=
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BAroundB[x1, x2, x3, x4, x5] /. d2 /. d1
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Out[4]=
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0
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