User:Jana/06-1350-HW4: Difference between revisions
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====R4==== |
====R4==== |
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This Reidemeister move has a number of forms. I will put two here. Pictures to come I hope. |
This Reidemeister move has a number of forms. I will put two here, both in linearized functional form. Pictures to come I hope. |
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R4c |
R4c |
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|<math>\rho_4c(x_1, x_2, x_3, x_4) = </math> |
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|<math>b^+(x_1,x_2,x_3) + \phi(x_1+x_2,x_3+x_4,x_4)</math> |
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|<math>- \phi+(x_1+x_2+x_3,x_3+x_4,x_4) - b^+(x_1,x_2,x_4) - b^+(x_1+x_4,x_3+x_4,x_4).</math> |
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Revision as of 21:53, 5 December 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
R4
This Reidemeister move has a number of forms. I will put two here, both in linearized functional form. Pictures to come I hope.
R4c
R4d
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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