0708-1300/Homework Assignment 8

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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 $K$ 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" presentation) with one generator for each edge of $D$ and two relations for each crossing of $D$, as indicated in the figure below.

Due Date

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