User:Jana/06-1350-HW4 - Revision history

Jana at 04:14, 6 December 2006

2006-12-06T04:14:16Z

← Older revision		Revision as of 00:14, 6 December 2006
Line 157:		Line 157:
	\|<math>- \rho_{4b}(x_1,x_3,x_4,x_5) + \rho_3(x_1+x_3,x_2,x_4,x_5).</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]]

Jana at 04:08, 6 December 2006

2006-12-06T04:08:53Z

@@ Line 27: / Line 27: @@
 ===The Relations===
 ====The Reidemeister Move R3====
@@ Line 44: / Line 56: @@
 ====R4====
-This Reidemeister move has a number of forms.  I will put two here, both in linearized functional form.
+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.
@@ Line 73: / Line 85: @@
 |<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>
 |}
 ===The Syzygies===
@@ Line 99: / 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>
 |}
 ===A Mathematica Verification===

Jana: /* The Generators */

2006-12-06T04:00:39Z

The Generators

← Older revision		Revision as of 00:00, 6 December 2006
Line 7:		Line 7:
	\|		\|
	\|		\|
			\|[[Image:Phi.PNG]]
	\|
	\|[[Image:06-1350-BPlus.svg\|100px]]		\|[[Image:06-1350-BPlus.svg\|100px]]
	\|		\|

Jana: /* R4 */

2006-12-06T03:56:03Z

← Older revision		Revision as of 23:56, 5 December 2006
Line 44:		Line 44:
	====R4====		====R4====

	This Reidemeister move has a number of forms. I will put two here, both in linearized functional form. ~~Pictures to come I hope.~~		This Reidemeister move has a number of forms. I will put two here, both in linearized functional form.


Line 65:		Line 65:

	[[Image:R4d.JPG]]		[[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>
			\|}

	===The Syzygies===		===The Syzygies===

Jana: /* R4 */

2006-12-06T03:55:06Z

← Older revision		Revision as of 23:55, 5 December 2006
Line 63:		Line 63:

	R4d		R4d

			[[Image:R4d.JPG]]

	===The Syzygies===		===The Syzygies===

Jana: /* R4 */

2006-12-06T03:52:46Z

← Older revision		Revision as of 23:52, 5 December 2006
Line 48:		Line 48:

	R4c		R4c

			[[Image:R4c.JPG]]


	{\| align=center		{\| align=center
	\|-		\|-

Jana: /* R4 */

2006-12-06T01:53:49Z

← Older revision		Revision as of 21:53, 5 December 2006
Line 54:		Line 54:
	\|-		\|-
	\|		\|
	\|<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>		\|<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



	===The Syzygies===		===The Syzygies===

Jana at 01:53, 6 December 2006

2006-12-06T01:53:14Z

← Older revision		Revision as of 21:53, 5 December 2006
Line 44:		Line 44:
	====R4====		====R4====

	This Reidemeister move has a number of forms. I will put two here. Pictures to come I hope.		This Reidemeister move has a number of forms. I will put two here, both in linearized functional form. Pictures to come I hope.


	R4c		R4c
			{\| 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>
			\|}

Jana at 01:42, 6 December 2006

2006-12-06T01:42:59Z

@@ Line 42: / Line 42: @@
 |<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===

Jana at 21:53, 23 November 2006

2006-11-23T21:53:34Z

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>}}