# HW7

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Study Apr 17 Office Hours
Exams Apr 24 Final, PM

Homework Assignment 7

required 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! Also, read Hatcher's section 1.3.

Just one problem. (due at the end of our class on Thursday, January 26, 2006). Let K be a knot in ${\mathbb R}^3$ presented by a planar diagram D. With a massive use of Van-Kampen's theorem, show that the fundamental group of the complement of K has a presentation (the "Wirtinger" [1] (http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Wirtinger.html) [2] (http://en.wikipedia.org/wiki/Wirtinger) presentation) with one generator for each edge of D and two relations for each crossing of D, as indicated in the figure below.

Just for fun. Prove that the following two links have homeomorphic fundamental groups (of their complements). In fact, show that their complements are actually homeomorphic. Are the links really the same?

(See more at [3] (http://www.math.toronto.edu/~drorbn/classes/0405/Topology/HW5/HW.html))

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