User:Jana/06-1350-HW4: Difference between revisions
No edit summary |
No edit summary |
||
(7 intermediate revisions by the same user not shown) | |||
Line 7: | Line 7: | ||
| |
| |
||
| |
| |
||
|[[Image:Phi.PNG]] |
|||
⚫ | |||
|[[Image:06-1350-BPlus.svg|100px]] |
|[[Image:06-1350-BPlus.svg|100px]] |
||
| |
| |
||
Line 27: | Line 27: | ||
===The Relations=== |
===The Relations=== |
||
====The Reidemeister Move R2 (Andy's)==== |
|||
The following version of R2 was the easiest to use to build my [[media:06-1350-PhiAroundPhi.png|original <math>\Phi</math> around <math>\Phi</math> syzygy]]: |
|||
[[Image:06-1350-R2-weird.png|center]] |
|||
In formulas, this is |
|||
<center><math>1 = (123)^\star B^- (132)^\star B^+.</math></center> |
|||
Linearized and written in functional form, this becomes |
|||
{| align=center |
|||
|- |
|||
|<math>\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==== |
====The Reidemeister Move R3==== |
||
Line 44: | Line 56: | ||
====R4==== |
====R4==== |
||
This Reidemeister move has a number of forms. I will put two here. |
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 |
R4c |
||
[[Image:R4c.JPG]] |
|||
{| align=center |
|||
|- |
|||
|<math>\rho_4c(x_1, x_2, x_3, x_4) = </math> |
|||
|<math>b^+(x_1,x_2,x_3) + \phi(x_1+x_2,x_3+x_4,x_4)</math> |
|||
|- |
|||
⚫ | |||
|<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> |
|||
|} |
|||
R4d |
R4d |
||
[[Image:R4d.JPG]] |
|||
{| align=center |
|||
|- |
|||
|<math>\rho_4d(x_1, x_2, x_3, x_4) = </math> |
|||
|<math>b^-(x_1,x_2,x_3) + \phi(x_1+x_2,x_3+x_4,x_4)</math> |
|||
|- |
|||
| |
|||
|<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> |
|||
|} |
|||
To establish the syzygy below, I needed two versions of R4. First: |
|||
[[Image:06-1350-R4a.png|center]] |
|||
In formulas, this is |
|||
<center><math>(1230)^\star B^+ (1213)^\star B^+ (1023)^\star \Phi = (1123)^\star \Phi (1233)^\star B^+</math>.</center> |
|||
Linearized and written in functional form, this becomes |
|||
{| align=center |
|||
|- |
|||
|<math>\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: |
|||
[[Image:06-1350-R4b.png|center]] |
|||
In formulas, this is |
|||
<center><math>(1123)^\star B^+ (1203)^\star B^+ (1231)^\star \Phi = (1230)^\star \Phi (1223)^\star B^+</math>.</center> |
|||
Linearized and written in functional form, this becomes |
|||
{| align=center |
|||
|- |
|||
|<math>\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> |
|||
|} |
|||
Are these independent, or can they be shown to be equivalent using other relations? |
|||
===The Syzygies=== |
===The Syzygies=== |
||
Line 80: | Line 133: | ||
|<math>+ \rho_3(x_1, x_3, x_4, x_5) + \rho_3(x_1 + x_3, x_2, x_4, x_5).</math> |
|<math>+ \rho_3(x_1, x_3, x_4, x_5) + \rho_3(x_1 + x_3, x_2, x_4, x_5).</math> |
||
|} |
|} |
||
====The "<math>\Phi</math> around B" Syzygy (By Andy)==== |
|||
The picture, with all shielding (and any other helpful notations) removed, is |
|||
{| align=center |
|||
|- align=center |
|||
|[[Image:06-1350-PhiAroundB.png|center]] |
|||
|- |
|||
|align=right|(Drawn with [http://asymptote.sf.net/ Asymptote], [[06-1350/Syzygies in Asymptote|Syzygies in Asymptote]]) |
|||
|} |
|||
The functional form of this syzygy is |
|||
{| align=center |
|||
|- |
|||
|<math>\Phi B(x_1,x_2,x_3,x_4,x_5) = </math> |
|||
|<math>\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)</math> |
|||
|- |
|||
| |
|||
|<math>- \rho_3(x_1,x_2,x_3+x_4,x_5) - \rho_{4a}(x_1,x_2,x_3,x_4)</math> |
|||
|- |
|||
| |
|||
|<math>- \rho_{4b}(x_1,x_3,x_4,x_5) + \rho_3(x_1+x_3,x_2,x_4,x_5).</math> |
|||
|} |
|||
====Phi around Phi==== |
|||
[[Image:Phiphi.svg]] |
|||
===A Mathematica Verification=== |
===A Mathematica Verification=== |
Latest revision as of 23:14, 5 December 2006
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
|