Bailey at 05:23, 6 December 2006

2006-12-06T05:23:48Z

← Older revision		Revision as of 01:23, 6 December 2006
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	Testing 123		Testing 123
			MORE TO COME

	===The Generators===		===The Generators===

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

			MORE TO COME

Bailey at 05:20, 6 December 2006

2006-12-06T05:20:34Z

← Older revision		Revision as of 01:20, 6 December 2006
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	Testing 123		Testing 123
			===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>}}

Bailey at 02:54, 6 December 2006

2006-12-06T02:54:10Z

New page

Testing 123

User:Bailey/HW4 - Revision history

Bailey at 05:23, 6 December 2006

Bailey at 05:20, 6 December 2006

Bailey at 02:54, 6 December 2006