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\centerline{\LARGE M.Sc.~Math~Workshop --- Assignment \#1}
\centerline{\large HUJI Spring 1998}
\centerline{Dror Bar-Natan}
\begin{enumerate}
\item
\begin{enumerate}
\item Prove that the fundamental group of the complement of a circle in
${\bold R}^3$ is $\bold Z$.
\item Prove that the fundamental group of the complement of a trefoil
knot is
\[ \left\langle
\alpha,\beta,\gamma:\,
\alpha=\beta^\gamma,\beta=\gamma^\alpha,\gamma=\alpha^\beta
\right\rangle, \]
where $a^b\eqdef b^{-1}ab$.
\begin{figure}[htpb]
\[ \eepic{WirtingerTrefoil}{0.5} \]
\caption{A hint.}
\end{figure}
\item Is this group Abelian? (Hint: think about maps into $S_3$)
\item Can you figure out a presentation for the fundamental group of an
arbitrary knot given in term of a planar projection?
\end{enumerate}
\item Prove: If $\lambda>0$ is irrational, $\epsilon>0$, and
$f:[0,1]\times[0,1]\to{\bold R}$ is continuous, then there exists
a continuous function $\phi:[0,1]\to[0,1]$ and a continuous function
$g:[0,1+\lambda]\to{\bold R}$ so that
$|f(x,y)-g(\phi(x)+\lambda\phi(y))|<\epsilon$ on at least 98\% of the
area of $[0,1]\times[0,1]$. (I.e., given addition, functions of two
variables ``almost factor'' through functions of one variable).
\item Can you find $2^{\aleph_0}$ different sets of natural numbers,
so that for any two of them $A$ and $B$, either $A\subset B$ or
$B\subset A$?
\item Let $G$ be a trivalent planar map (that is, a trivalent graph
embedded in the plane). Prove that there are as many edge 3-colorings of
$G$ (colorings of the edges of $G$ by 3 colors, so that the 3 edges
meeting at any given vertex are of different colors) as face 4-colorings of
$G$ (that is, map 4-colorings) in which ``the state at infinity'' is
blue.
\item Prove: The unit sphere in an infinite-dimensional Hilbert space
is contractible.
\item What is the configuration space of the machine $M_2$ in the picture
below?
\begin{figure}[htpb]
\[ \eepic{M2}{0.75} \]
\caption{$M_2$ is a planar machine (a machine constrained to move in the
plane) made of 4 rods, three joints, and two bolts arranged as above.
The rods are straight and inflexible. The joints allow perfect bending,
and the bolts are fixed to some fixed points in the plane. The relative
sizes and distances are as shown.
}
\end{figure}
\item Prove that the group of symmetries of the dodecahedron is $A_5$.
\end{enumerate}
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