User:Jana/061350HW4
The Generators
Our generators are , , and :
Picture  
Generator  
Perturbation 
The Relations
The Reidemeister Move R2 (Andy's)
The following version of R2 was the easiest to use to build my original around syzygy:
In formulas, this is
Linearized and written in functional form, this becomes
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. The two following were copied from Andy.
R4c
R4d
To establish the syzygy below, I needed two versions of R4. First:
In formulas, this is
Linearized and written in functional form, this becomes
Second:
In formulas, this is
Linearized and written in functional form, this becomes
Are these independent, or can they be shown to be equivalent using other relations?
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
The " around B" Syzygy (By Andy)
The picture, with all shielding (and any other helpful notations) removed, is
(Drawn with Asymptote, Syzygies in Asymptote) 
The functional form of this syzygy is
Phi around Phi
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
