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

Andy at 22:35, 5 December 2006

2006-12-05T22:35:42Z

← Older revision		Revision as of 18:35, 5 December 2006
Line 29:		Line 29:

	====The Reidemeister Move R2====		====The Reidemeister Move R2====
	The following version of R2 was the easiest to use to ~~program~~ ~~the~~ <math>\Phi</math> around <math>\Phi</math> syzygy:		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]]		[[Image:06-1350-R2-weird.png\|center]]
	In formulas, this is		In formulas, this is
Line 129:		Line 129:
	====The "<math>\Phi</math> around <math>\Phi</math>" Syzygy====		====The "<math>\Phi</math> around <math>\Phi</math>" Syzygy====

	The original ~~picture~~ is available at [[:Image:06-1350-PhiAroundPhi.png]].		The original syzygy is available at [[:Image:06-1350-PhiAroundPhi.png]].
	A cleaner, minimal picture is		A cleaner, minimal picture is
	{\| align=center		{\| align=center
Line 140:		Line 140:
	The functional form of this syzygy is		The functional form of this syzygy is

			{\|align=center
	{\|
	\|-		\|-
	\|<math>\Phi\Phi(x_1,x_2,x_3,x_4,x_5) = </math>		\|<math>\Phi\Phi(x_1,x_2,x_3,x_4,x_5) = </math>

Andy: Added cleaner Phi around Phi syzygy.

2006-12-05T22:30:30Z

Added cleaner Phi around Phi syzygy.

← Older revision		Revision as of 18:30, 5 December 2006
Line 129:		Line 129:
	====The "<math>\Phi</math> around <math>\Phi</math>" Syzygy====		====The "<math>\Phi</math> around <math>\Phi</math>" Syzygy====

			The original picture is available at [[:Image:06-1350-PhiAroundPhi.png]].
	The picture is
			A cleaner, minimal picture is
	{\| align=center		{\| align=center
	\|- align=center		\|- align=center
	\|[[Image:06-1350-~~PhiAroundPhi~~.png\|center]]		\|[[Image:06-1350-PhiAroundPhiClean.png\|center]]
	\|-		\|-
	\|align=right\|(Drawn with [http://asymptote.sf.net/ Asymptote], [[06-1350/Syzygies in Asymptote\|Syzygies in Asymptote]])		\|align=right\|(Drawn with [http://asymptote.sf.net/ Asymptote], [[06-1350/Syzygies in Asymptote\|Syzygies in Asymptote]])
Line 139:		Line 140:
	The functional form of this syzygy is		The functional form of this syzygy is

		⚫	{\|
	{\| align=center
	\|-		\|-
	\|<math>\Phi\Phi(x_1,x_2,x_3,x_4,x_5) = </math>		\|<math>\Phi\Phi(x_1,x_2,x_3,x_4,x_5) = </math>
	\|<math>\~~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~~)</math>		\|<math>\rho_{4b}(x_1+x_4,x_2,x_3,x_5) + \rho_{4b}(x_1,x_2,x_3,x_4) + \rho_{4a}(x_1,x_2+x_3,x_4,x_5)</math>
	\|-		\|-
	\|		\|
	\|<math>+ \~~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~~)</math>		\|<math>- \rho_{4b}(x_1,x_2,x_3,x_4+x_5) - \rho_{4a}(x_1+x_2,x_3,x_4,x_5) - \rho_{4a}(x_1,x_2,x_4,x_5).</math>
	\|-
⚫	\|
	\|<math>+ \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)</math>
	\|-
	\|
	\|<math>- \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)</math>
	\|-
	\|
	\|<math>- \rho_2(x_1,x_2,x_4) - \rho_2(x_1+x_2,x_3,x_4)</math>
	\|}		\|}

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

	===A Mathematica Verification===		===A Mathematica Verification===

Andy: /* The " $\Phi$ around $\Phi$ " Syzygy */

2006-12-05T21:53:41Z

The "<math>\Phi</math> around <math>\Phi</math>" Syzygy

← Older revision		Revision as of 17:53, 5 December 2006
Line 157:		Line 157:
	\|}		\|}

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

	===A Mathematica Verification===		===A Mathematica Verification===

Andy: Added weird R2.

2006-12-05T19:30:23Z

Added weird R2.

← Older revision		Revision as of 15:30, 5 December 2006
Line 27:		Line 27:

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

			====The Reidemeister Move R2====
			The following version of R2 was the easiest to use to program the <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====

Andy: Added Phi around Phi syzygy.

2006-12-05T19:14:36Z

Added Phi around Phi syzygy.

@@ Line 115: / Line 115: @@
 |<math>- \rho_{4b}(x_1,x_3,x_4,x_5) + \rho_3(x_1+x_3,x_2,x_4,x_5).</math>
 |}
 ===A Mathematica Verification===

128.100.68.3: Added link.

2006-12-04T19:29:14Z

Added link.

← Older revision		Revision as of 15:29, 4 December 2006
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	\|[[Image:06-1350-PhiAroundB.png\|center]]		\|[[Image:06-1350-PhiAroundB.png\|center]]
	\|-		\|-
	\|align=right\|(Drawn with [http://asymptote.sf.net/ Asymptote])		\|align=right\|(Drawn with [http://asymptote.sf.net/ Asymptote], [[06-1350/Syzygies in Asymptote\|Syzygies in Asymptote]])
	\|}		\|}

Andy: Added equation for syzygy.

2006-12-03T23:20:22Z

Added equation for syzygy.

← Older revision		Revision as of 19:20, 3 December 2006
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	\|}		\|}

	====The Reidemeister Move R3====		====The Reidemeister Move R4====
	To establish the syzygy below, I needed two versions of R4. First:		To establish the syzygy below, I needed two versions of R4. First:
	[[Image:06-1350-R4a.png\|center]]		[[Image:06-1350-R4a.png\|center]]
Line 100:		Line 100:
	\|-		\|-
	\|align=right\|(Drawn with [http://asymptote.sf.net/ Asymptote])		\|align=right\|(Drawn with [http://asymptote.sf.net/ 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>
	\|}		\|}

Andy: Adding R4 relations.

2006-12-03T23:12:30Z

Adding R4 relations.

@@ 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===

Andy: Added Phi around B picture.

2006-12-03T17:11:31Z

Added Phi around B picture.

@@ Line 67: / Line 67: @@
+|
 |<math>+ \rho_3(x_1, x_3, x_4, x_5) + \rho_3(x_1 + x_3, x_2, x_4, x_5).</math>
 |}

Andy: Copied the assignment.

2006-11-27T21:51:41Z

Copied the assignment.

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>}}