User:Zsuzsi/HW4 - Revision history

Zsuzsi at 23:48, 5 December 2006

2006-12-05T23:48:57Z

@@ Line 52: / Line 52: @@
 ====The symmetry of <math>\Phi</math>====
-<math>\Phi</math> has to have A(4)-symmetry. A(4) is generated by 120 degree rotations around the vertices of the tetrahedron. For example, rotation around the "top" vertex yields the following picture and relation:
+<math>\Phi</math> has to have A(4)-symmetry. For example, rotation around the "top" vertex yields the following picture and relation:
 [[Image:Symm3.jpg]]
@@ Line 60: / Line 60: @@
 <math>s_3(x_1,x_2,x_3)=\varphi(x_1,x_2,x_3)-\varphi(-x_1-x_2,-x_2-x_3,x_2)</math>
 ====The Reidemeister move R1====

128.100.68.3 at 22:28, 5 December 2006

2006-12-05T22:28:33Z

@@ Line 84: / Line 84: @@
 <math>\rho_2(x_1,x_2,x_3)=b^+(x_1,x_2,x_3)+b^-(x_1,x_3,x_2) </math>
@@ Line 171: / Line 175: @@
 |<math>- \rho_{4b}(x_1,x_3,x_4,x_5) + \rho_3(x_1+x_3,x_2,x_4,x_5).</math>
 |}

Zsuzsi at 18:55, 5 December 2006

2006-12-05T18:55:56Z

@@ Line 147: / Line 147: @@
 |<math>+ \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===

Zsuzsi at 18:51, 5 December 2006

2006-12-05T18:51:00Z

@@ Line 31: / Line 31: @@
 ===The Relations===
 ====The Reidemeister move R1====
@@ Line 36: / Line 65: @@
 [[Image:Reidemeister1.jpg]]
-In the linearized functional notation, if I'm getting this right, it looks like this:
+In the linearized functional notation, it looks like this:
 <math> \rho_1(x_1,x_2)=b^-(x_1,x_2,-x_2)</math>
@@ Line 56: / Line 85: @@
 <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====

Zsuzsi at 21:43, 4 December 2006

2006-12-04T21:43:44Z

@@ Line 36: / Line 36: @@
 [[Image:Reidemeister1.jpg]]
 ====The Reidemeister move R2====
 [[Image:Reidemeister2.jpg]]
 ====The Reidemeister Move R3====
@@ Line 54: / Line 71: @@
+|
 |<math>- 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>
 |}

Zsuzsi at 21:15, 4 December 2006

2006-12-04T21:15:43Z

← Older revision		Revision as of 17:15, 4 December 2006
Line 34:		Line 34:
	====The Reidemeister move R1====		====The Reidemeister move R1====

	[[Image:~~reidemeister1~~]]		[[Image:Reidemeister1.jpg]]


	====The Reidemeister move R2====		====The Reidemeister move R2====

	[[Image:~~reidemeister2~~]]		[[Image:Reidemeister2.jpg]]

	====The Reidemeister Move R3====		====The Reidemeister Move R3====

Zsuzsi at 21:13, 4 December 2006

2006-12-04T21:13:51Z

← Older revision		Revision as of 17:13, 4 December 2006
Line 25:		Line 25:
	\|<math>b^-</math>		\|<math>b^-</math>
	\|}		\|}

			A low-tech completed version of this chart:

			[[Image:chart.jpg]]

	===The Relations===		===The Relations===

			====The Reidemeister move R1====

			[[Image:reidemeister1]]


			====The Reidemeister move R2====

			[[Image:reidemeister2]]

	====The Reidemeister Move R3====		====The Reidemeister Move R3====

Zsuzsi at 18:44, 4 December 2006

2006-12-04T18:44:23Z

New page

===The Generators===

Our generators are <math>T</math>, <math>R</math>, <math>\Phi</math> and <math>B^{\pm}</math>:
{| align=center cellpadding=10 style="border: solid orange 1px"
|- align=center valign=middle
|align=left|Picture
|
|
|
|[[Image:06-1350-BPlus.svg|100px]]
|
|- align=center valign=middle
|align=left|Generator
|<math>T</math>
|<math>R</math>
|<math>\Phi</math>
|<math>B^+</math>
|<math>B^-</math>
|- align=center valign=middle
|align=left|Perturbation
|<math>t</math>
|<math>r</math>
|<math>\varphi</math>
|<math>b^+</math>
|<math>b^-</math>
|}

===The Relations===

====The Reidemeister Move R3====
The picture (with three sides of the shielding removed) is
[[Image:06-1350-R4.svg|400px|center]]
In formulas, this is
<center><math>(1230)^\star B^+ (1213)^\star B^+ (1023)^\star B^+ = (1123)^\star B^+ (1203)^\star B^+ (1231)^\star B^+</math>.</center>
Linearized and written in functional form, this becomes
{| align=center
|-
|<math>\rho_3(x_1, x_2, x_3, x_4) = </math>
|<math>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>- 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 Syzygies===

====The "B around B" Syzygy====

The picture, with all shielding removed, is
{| align=center
|- align=center
|[[Image:06-1350-BAroundB.svg|center]]
|-
|align=right|(Drawn with [http://www.inkscape.org/ Inkscape])<br>(note that lower quality pictures are also acceptable)
|}

The functional form of this syzygy is

{| align=center
|-
|<math>BB(x_1,x_2,x_3,x_4,x_5) = </math>
|<math>\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>- \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>+ \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>d^2=0</math>. Copy paste it into a live Mathematica session to see that it's right!

{{In|n=1|in=<nowiki>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]
};</nowiki>}}

{{InOut|n=3|in=<nowiki>BAroundB[x1, x2, x3, x4, x5] /. d2</nowiki>|out=<nowiki>- 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]</nowiki>}}

{{InOut|n=4|in=<nowiki>BAroundB[x1, x2, x3, x4, x5] /. d2 /. d1</nowiki>|out=<nowiki>0</nowiki>}}