User:Sankaran/06-1350-HW4

From Drorbn
Jump to navigationJump to search


The Generators

Our generators are [math]\displaystyle{ T }[/math], [math]\displaystyle{ R }[/math], [math]\displaystyle{ \Phi }[/math] and [math]\displaystyle{ B^{\pm} }[/math]:

Picture 06-1350-T.svg 06-1350-R.svg 06-1350-Phi.svg 06-1350-BPlus.svg 06-1350-BMinus.svg
Generator [math]\displaystyle{ T }[/math] [math]\displaystyle{ R }[/math] [math]\displaystyle{ \Phi }[/math] [math]\displaystyle{ B^+ }[/math] [math]\displaystyle{ B^- }[/math]
Perturbation [math]\displaystyle{ t }[/math] [math]\displaystyle{ r }[/math] [math]\displaystyle{ \varphi }[/math] [math]\displaystyle{ b^+ }[/math] [math]\displaystyle{ b^- }[/math]

(Thanks Zavosh for the nice picture)

The Relations

The Reidemeister Move R2

(Courtesy of Andy)

06-1350-R2-weird.png

In formulas, this is

[math]\displaystyle{ 1 = (123)^\star B^- (132)^\star B^+. }[/math]

Linearized and written in functional form, this becomes

[math]\displaystyle{ \rho_2(x_1,x_2,x_3) = - b^-(x_1,x_2,x_3) - b^+(x_1,x_3,x_2). }[/math]

The Reidemeister Move R3

(Picture and first example courtesy of Dror)

There are eight of these (each crossing in the picture can be + or - ). For example, if all the crossings are positive, the picture (with three sides of the shielding removed) is

06-1350-R4.svg

In formulas, this is

[math]\displaystyle{ (1230)^\star B^+ (1213)^\star B^+ (1023)^\star B^+ = (1123)^\star B^+ (1203)^\star B^+ (1231)^\star B^+ }[/math].

Linearized and written in functional form, this becomes

[math]\displaystyle{ \rho_3[+++](x_1, x_2, x_3, x_4) = }[/math] [math]\displaystyle{ b^+(x_1,x_2,x_3) + b^+(x_1+x_3,x_2,x_4) + b^+(x_1,x_3,x_4) }[/math]
[math]\displaystyle{ - 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). }[/math]

Here are the rest of them, linearized and in functional form - I think this is too many, but it's probably easier to write these out than to figure the relationships between them. Also, some better notation is needed.

[math]\displaystyle{ \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). }[/math]

[math]\displaystyle{ \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). }[/math]

[math]\displaystyle{ \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). }[/math]

[math]\displaystyle{ \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). }[/math]

[math]\displaystyle{ \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). }[/math]

[math]\displaystyle{ \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). }[/math]

[math]\displaystyle{ \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). }[/math]

The Reidemeister Move R4

(Courtesy of Andy)

There are two (ostensibly) different versions:

06-1350-R4a.png

In formulas, this is

[math]\displaystyle{ (1230)^\star B^+ (1213)^\star B^+ (1023)^\star \Phi = (1123)^\star \Phi (1233)^\star B^+ }[/math].

Linearized and written in functional form, this becomes

[math]\displaystyle{ \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). }[/math]

Second:

06-1350-R4b.png

In formulas, this is

[math]\displaystyle{ (1123)^\star B^+ (1203)^\star B^+ (1231)^\star \Phi = (1230)^\star \Phi (1223)^\star B^+ }[/math].

Linearized and written in functional form, this becomes

[math]\displaystyle{ \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). }[/math]


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

[math]\displaystyle{ BB(x_1,x_2,x_3,x_4,x_5) = }[/math] [math]\displaystyle{ \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]
[math]\displaystyle{ - \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]
[math]\displaystyle{ + \rho_3(x_1, x_3, x_4, x_5) + \rho_3(x_1 + x_3, x_2, x_4, x_5). }[/math]

A Mathematica Verification

The following simulated Mathematica session proves that for our single relation and single syzygy, [math]\displaystyle{ d^2=0 }[/math]. 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
Retrieved from "https://drorbn.net/index.php?title=User:Sankaran/06-1350-HW4&oldid=3264"