0708-1300/Class notes for Thursday, January 17 - Revision history

Trefor at 23:16, 6 February 2008

2008-02-06T23:16:08Z

← Older revision		Revision as of 19:16, 6 February 2008
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	Let X be a point pointed topological space such that <math>X = U_1\cup U_2</math> where <math>U_1</math> and <math>U_2</math> are open and the base point b is in the (connected) intersection.		Let X be a point pointed topological space such that <math>X = U_1\cup U_2</math> where <math>U_1</math> and <math>U_2</math> are open and the base point b is in the (connected) intersection.

	Then, <math>\pi_1() = \pi_1(U_1)*_{\pi_1(U_1\cap U_2)}\pi_1(U_2)</math>		Then, <math>\pi_1(X) = \pi_1(U_1)*_{\pi_1(U_1\cap U_2)}\pi_1(U_2)</math>

Canghel at 04:53, 6 February 2008

2008-02-06T04:53:11Z

← Older revision		Revision as of 00:53, 6 February 2008
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	<math>\pi_1^{ab}(\Sigma_2) = \pi_1(\Sigma_2)/ gh=hg = F.A.G (\alpha_1,\alpha_2,\beta_1,\beta^2) = \mathbb{Z}^4 \neq \mathbb{Z}^2</math>		<math>\pi_1^{ab}(\Sigma_2) = \pi_1(\Sigma_2)/ gh=hg = F.A.G (\alpha_1,\alpha_2,\beta_1,\beta^2) = \mathbb{Z}^4 \neq \mathbb{Z}^2</math>


			In case someone might want diagrams for the examples above:

			[[Image:0708-1300_notes_17-01-08c.jpg\|200px]]

Trefor at 04:17, 22 January 2008

2008-01-22T04:17:47Z

Show changes

Drorbn at 22:28, 21 January 2008

2008-01-21T22:28:37Z

← Older revision		Revision as of 18:28, 21 January 2008
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	{{0708-1300/Navigation}}		{{0708-1300/Navigation}}

	==Pictures for a Van-Kampen Computation==
	{{In\|
	n = 1 \|
	in = <nowiki><< KnotTheory`</nowiki>}}

	<tt>Loading KnotTheory` version of January 13, 2008, 20:30:12.1353.<br>
	Read more at http://katlas.org/wiki/KnotTheory.</tt>

	{{Graphics\|
	n = 2 \|
	in = <nowiki>TubePlot[TorusKnot[8, 3]]</nowiki> \|
	img= 0708-1300-T83.png}}

	{{In\|
	n = 3 \|
	in = <nowiki>TC[r1_, t1_,r2_,t2_ ] := {
	(r1 +r2 Cos[2Pi t2])Cos[2Pi t1],
	(r1 +r2 Cos[2Pi t2])Sin[2Pi t1],
	r2 Sin[2Pi t2]
	};</nowiki>}}

	{{In\|
	n = 4 \|
	in = <nowiki>InflatedTorus[p_, q_, b_] := ParametricPlot3D[
	TC[
	2, p t - q s,
	1 + b(p^2 + q^2)s(1 - (p^2 + q^2)s), q t + p s
	],
	{t, 0, 1}, {s, 0, 1/(p^2 + q^2)},
	PlotPoints -> {6(p^2 + q^2) + 1, 7},
	DisplayFunction -> Identity
	];</nowiki>}}

	{{Graphics\|
	n = 5 \|
	in = <nowiki>GraphicsArray[{{InflatedTorus[3,8,1], InflatedTorus[3,8,-1]}}]</nowiki> \|
	img= 0708-1300-InflatedTori.png \|
	width = 640px}}

Drorbn at 14:55, 17 January 2008

2008-01-17T14:55:43Z

← Older revision		Revision as of 10:55, 17 January 2008
Line 38:		Line 38:
	in = <nowiki>GraphicsArray[{{InflatedTorus[3,8,1], InflatedTorus[3,8,-1]}}]</nowiki> \|		in = <nowiki>GraphicsArray[{{InflatedTorus[3,8,1], InflatedTorus[3,8,-1]}}]</nowiki> \|
	img= 0708-1300-InflatedTori.png \|		img= 0708-1300-InflatedTori.png \|
	width = ~~720px~~}}		width = 640px}}

Drorbn at 14:53, 17 January 2008

2008-01-17T14:53:51Z

← Older revision		Revision as of 10:53, 17 January 2008
Line 12:		Line 12:
	n = 2 \|		n = 2 \|
	in = <nowiki>TubePlot[TorusKnot[8, 3]]</nowiki> \|		in = <nowiki>TubePlot[TorusKnot[8, 3]]</nowiki> \|
	img= 0708-1300-T83.png \|		img= 0708-1300-T83.png}}
	out= <nowiki>-Graphics-</nowiki>}}

	{{In\|		{{In\|
Line 39:		Line 38:
	in = <nowiki>GraphicsArray[{{InflatedTorus[3,8,1], InflatedTorus[3,8,-1]}}]</nowiki> \|		in = <nowiki>GraphicsArray[{{InflatedTorus[3,8,1], InflatedTorus[3,8,-1]}}]</nowiki> \|
	img= 0708-1300-InflatedTori.png \|		img= 0708-1300-InflatedTori.png \|
			width = 720px}}
	out= <nowiki>-GraphicsArray-</nowiki>}}

Drorbn at 14:48, 17 January 2008

2008-01-17T14:48:08Z

← Older revision		Revision as of 10:48, 17 January 2008
Line 1:		Line 1:
	{{0708-1300/Navigation}}		{{0708-1300/Navigation}}

			==Pictures for a Van-Kampen Computation==
	{{In\|		{{In\|
	n = 1 \|		n = 1 \|

Drorbn at 14:46, 17 January 2008

2008-01-17T14:46:52Z

New page

{{0708-1300/Navigation}}

{{In|
n = 1 |
in = <nowiki><< KnotTheory`</nowiki>}}

<tt>Loading KnotTheory` version of January 13, 2008, 20:30:12.1353.<br>
Read more at http://katlas.org/wiki/KnotTheory.</tt>

{{Graphics|
n = 2 |
in = <nowiki>TubePlot[TorusKnot[8, 3]]</nowiki> |
img= 0708-1300-T83.png |
out= <nowiki>-Graphics-</nowiki>}}

{{In|
n = 3 |
in = <nowiki>TC[r1_, t1_,r2_,t2_ ] := {
(r1 +r2 Cos[2Pi t2])Cos[2Pi t1],
(r1 +r2 Cos[2Pi t2])Sin[2Pi t1],
r2 Sin[2Pi t2]
};</nowiki>}}

{{In|
n = 4 |
in = <nowiki>InflatedTorus[p_, q_, b_] := ParametricPlot3D[
TC[
2, p t - q s,
1 + b(p^2 + q^2)s(1 - (p^2 + q^2)s), q t + p s
],
{t, 0, 1}, {s, 0, 1/(p^2 + q^2)},
PlotPoints -> {6(p^2 + q^2) + 1, 7},
DisplayFunction -> Identity
];</nowiki>}}

{{Graphics|
n = 5 |
in = <nowiki>GraphicsArray[{{InflatedTorus[3,8,1], InflatedTorus[3,8,-1]}}]</nowiki> |
img= 0708-1300-InflatedTori.png |
out= <nowiki>-GraphicsArray-</nowiki>}}