0708-1300/Homework Assignment 8 - Revision history

Drorbn at 14:51, 24 January 2008

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	{{0708-1300/Navigation}}		{{0708-1300/Navigation}}
	{{In Preparation}}

	==Reading==		==Reading==

Drorbn at 14:49, 24 January 2008

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	==Doing==		==Doing==
	'''Just one problem.''' Let <math>K</math> be a knot in <math>{\mathbb R}^3</math> presented by a planar diagram <math>D</math>. With a massive use of Van-Kampen's theorem, show that the fundamental group of the complement of <math>K</math> has a presentation (the "Wirtinger" ~~[http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Wirtinger.html] [http://en.wikipedia.org/wiki/Wirtinger]~~ presentation) with one generator for each edge of <math>D</math> and two relations for each crossing of <math>D</math>, as indicated in the figure below.		'''Just one problem.''' Let <math>K</math> be a knot in <math>{\mathbb R}^3</math> presented by a planar diagram <math>D</math>. With a massive use of Van-Kampen's theorem, show that the fundamental group of the complement of <math>K</math> has a presentation (the "Wirtinger" presentation) with one generator for each edge of <math>D</math> and two relations for each crossing of <math>D</math>, as indicated in the figure below.

	<center>		<center>

Drorbn at 14:44, 24 January 2008

2008-01-24T14:44:07Z

New page

{{0708-1300/Navigation}}
{{In Preparation}}

==Reading==
Read, reread and rereread your notes to this point, and make sure that you really, really really, really really really understand everything in them. Do the same every week!

==Doing==
'''Just one problem.''' Let <math>K</math> be a knot in <math>{\mathbb R}^3</math> presented by a planar diagram <math>D</math>. With a massive use of Van-Kampen's theorem, show that the fundamental group of the complement of <math>K</math> has a presentation (the "Wirtinger" [http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Wirtinger.html] [http://en.wikipedia.org/wiki/Wirtinger] presentation) with one generator for each edge of <math>D</math> and two relations for each crossing of <math>D</math>, as indicated in the figure below.

<center>
[[Image:0708-1300-WirtingerExample.png]]
</center>

==Due Date==
This assignment is due in class on Thursday February 7, 2008.