User:Andy/06-1350-HW4

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Contents

The Generators

Our generators are T, R, \Phi and B^{\pm}:

Picture 06-1350-BPlus.svg
Generator T R \Phi B^+ B^-
Perturbation t r \varphi b^+ b^-

The Relations

The Reidemeister Move R2

The following version of R2 was the easiest to use to program the \Phi around \Phi syzygy:

06-1350-R2-weird.png

In formulas, this is

1 = (123)^\star B^- (132)^\star B^+.

Linearized and written in functional form, this becomes

\rho_2(x_1,x_2,x_3) =  - b^-(x_1,x_2,x_3) - b^+(x_1,x_3,x_2).

The Reidemeister Move R3

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

06-1350-R4.svg

In formulas, this is

(1230)^\star B^+ (1213)^\star B^+ (1023)^\star B^+ = (1123)^\star B^+ (1203)^\star B^+ (1231)^\star B^+.

Linearized and written in functional form, this becomes

\rho_3(x_1, x_2, x_3, x_4) = b^+(x_1,x_2,x_3) + b^+(x_1+x_3,x_2,x_4) + b^+(x_1,x_3,x_4)
- b^+(x_1+x_2,x_3,x_4) - b^+(x_1,x_2,x_4) - b^+(x_1+x_4,x_2,x_3).

The Reidemeister Move R4

To establish the syzygy below, I needed two versions of R4. First:

06-1350-R4a.png

In formulas, this is

(1230)^\star B^+ (1213)^\star B^+ (1023)^\star \Phi = (1123)^\star \Phi (1233)^\star B^+.

Linearized and written in functional form, this becomes

\rho_{4a}(x_1,x_2,x_3,x_4) = b^+(x_1,x_2,x_3) + b^+(x_1+x_3,x_2,x_4) + \phi(x_1,x_3,x_4) - \phi(x_1+x_2,x_3,x_4) - b^+(x_1,x_2,x_3+x_4).

Second:

06-1350-R4b.png

In formulas, this is

(1123)^\star B^+ (1203)^\star B^+ (1231)^\star \Phi = (1230)^\star \Phi (1223)^\star B^+.

Linearized and written in functional form, this becomes

\rho_{4b}(x_1,x_2,x_3,x_4) = b^+(x_1+x_2,x_3,x_4) + b^+(x_1,x_2,x_4) + \phi(x_1+x_4,x_2,x_3) - \phi(x_1,x_2,x_3) - b^+(x_1,x_2+x_3,x_4).

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

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

The functional form of this syzygy is

BB(x_1,x_2,x_3,x_4,x_5) = \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)
- \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)
+ \rho_3(x_1, x_3, x_4, x_5) + \rho_3(x_1 + x_3, x_2, x_4, x_5).

The "\Phi around B" Syzygy

The picture, with all shielding (and any other helpful notations) removed, is

06-1350-PhiAroundB.png
(Drawn with Asymptote, Syzygies in Asymptote)

The functional form of this syzygy is

\Phi B(x_1,x_2,x_3,x_4,x_5) = \rho_3(x_1,x_2,x_3,x_5) + \rho_{4a}(x_1+x_5,x_2,x_3,x_4) + \rho_{4b}(x_1+x_2,x_3,x_4,x_5)
- \rho_3(x_1,x_2,x_3+x_4,x_5) - \rho_{4a}(x_1,x_2,x_3,x_4)
- \rho_{4b}(x_1,x_3,x_4,x_5) + \rho_3(x_1+x_3,x_2,x_4,x_5).

The "\Phi around \Phi" Syzygy

The picture is

06-1350-PhiAroundPhi.png
(Drawn with Asymptote, Syzygies in Asymptote)

The functional form of this syzygy is

\Phi\Phi(x_1,x_2,x_3,x_4,x_5) = \rho_2(x_1+x_2,x_3,x_4) + \rho_2(x_1+x_2+x_4,x_3,x_5) + \rho_{4b}(x_1+x_2,x_4,x_5,x_3)
+ \rho_2(x_1,x_2,x_4) + \rho_2(x_1+x_4,x_2,x_5) + \rho_{4b}(x_1,x_4,x_5,x_2)
+ \rho_{4a}(x_1,x_4+x_5,x_2,x_3) - \rho_{4b}(x_1,x_4,x_5,x_2+x_3) - \rho_{4a}(x_1+x_4,x_5,x_2,x_3)
- \rho_2(x_1+x_4,x_2,x_5) - \rho_2(x_1+x_2+x_4,x_3,x_5) - \rho_{4a}(x_1,x_4,x_2,x_3)
- \rho_2(x_1,x_2,x_4) - \rho_2(x_1+x_2,x_3,x_4)

Note that the first and last terms cancel, as the top steps at the top of the diagram are opposites.

A Mathematica Verification

The following simulated Mathematica session proves that for our single relation and single syzygy, d^2=0. 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