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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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