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\title{Math 139 Final Exam}

\author{Danny Calegari}

\maketitle
Answer all problems. These problems are all of equal weight. If you have questions
or comments, contact me by email at {\tt dannyc@math.harvard.edu}.
Solutions to this exam are due in my mailbox at 3pm on Tuesday, May 15th.

\begin{prob}
How would you define the Dehn invariant for three dimensional hyperbolic polyhedra?
What about for finite volume polyhedra possibly with ideal vertices?
Find an infinite family of three dimensional hyperbolic polyhedra with volume
uniformly bounded above and Dehn invariant $0$.
\end{prob}

\begin{prob}
Fix a point $p \in \H^3$ and consider a sequence of points $q_i,r_i \in \H^3$
such that the distance $d(p,q_i) \to \infty$ and $d(p,r_i) \to \infty$.
Show that the angles at $p$ of the hyperbolic triangles $q_ipr_i$ converge to
$0$ if and only if the distances
$$d(p,q_i) + d(p,r_i) - d(q_i,r_i) \to \infty$$
\end{prob}

\begin{prob}
This problem involves the calculation of a volume of a certain hyperbolic
tetrahedron.
\begin{enumerate}
\item{Let $$A(x) = \int_0^x \frac 1 {1+t^2} \log \frac 4 {1+t^2} dt$$
for $t \in \R$. Show that $A(x) = 2\Lambda(\cot^{-1} x)$
where $\Lambda$ is Lobachevsky's function.}
\item{Let $G_{\alpha,\gamma}$ be the hyperbolic
tetrahedron with one ideal vertex defined as follows: let the
vertices of $T$ be $p,q,r,s$ with $p$ an ideal point. Suppose that
the triangle $pqr$ and $pqs$ have right angles at $q$. Suppose that
the dihedral angle along edge $pq$ is $\alpha$ and along $rs$ is $\gamma$, and
the dihedral angles along $qr$,$qs$ and $pr$ are $\pi/2$. Show that
$G_{\alpha,\gamma}$ is uniquely defined up to isometry by these conditions.}
\item{Show that the volume of $G_{\alpha,\gamma}$ is given by
$$\text{vol}(G_{\alpha,\gamma}) = {\frac 1 8} \Bigl( A\Bigl(\frac {1-ac} {a+c}
\Bigr) + A\Bigl(\frac {1+ac} {a-c} \Bigr) + 2 A(a)\Bigr)$$
where $a = \tan(\alpha)$ and $c=\tan(\gamma)$.}
\end{enumerate}
\end{prob}

\begin{prob}
We say a knot $K$ in $\S^3$ has {\em tunnel number $1$} if
there is an arc $\alpha$ in $\S^3 - K$ with both endpoints on $K$, such
that $\S^3 - N(K \cup \alpha)$ is a genus $2$ handlebody (here $N$ denotes
a small thickened neighborhood of the graph $K \cup \alpha$).
\begin{enumerate}
\item{Show that if $K$ has tunnel number $1$,
there is a presentation for $\pi_1(\S^3-K)$ with two generators and one relation.}
\item{Show that the figure eight knot has tunnel number $1$.}
\end{enumerate}
\end{prob}

\begin{prob}
Give three examples of knots whose complements do not admit complete finite
volume hyperbolic structures, for three different reasons.

Give an example of a knot --- {\em other than the figure eight knot!}  --- whose
complement in $\S^3$ admits a complete finite volume hyperbolic structure.
Construct such a structure by finding a decomposition of the complement into
ideal hyperbolic tetrahedra, and find a solution to the edge and completeness
equations with all simplices positively oriented.
\end{prob}

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