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The Final Exam

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Solve and submit your solution of two (just two!) of the following three questions by noon on Tuesday January 6, 2004. Remember -- Elegance counts!!! If you can type your solution, that's better. If you can't, at least copy it again to a clean sheet of paper. Formulas without words explaining them will not be accepted!

  1. Prove in detail:
    1. All torus knots, except for the obvious exceptions, are really knotted.
    2. All knotted torus knots are prime.

  2. The ``Dubrovnik Polynomial'' $ D$ (a variant of the ``Kauffman Polynomial'' $ L$) is an invariant of framed links valued in rational functions in the variables $ a$ and $ z$, satisfying the following relations:
    $\displaystyle D(\bigcirc)$ $\displaystyle =$ $\displaystyle 1,$ (1)
    $\displaystyle D({\hspace{-3pt}\begin{array}{c}\raisebox{-2pt}{\includegraphics[width=8mm]{figs/Rkink.eps}}\end{array}\hspace{-3pt}})$ $\displaystyle =$ $\displaystyle aD({\hspace{-3pt}\begin{array}{c}\raisebox{1pt}{\includegraphics[width=8mm]{figs/Nkink.eps}}\end{array}\hspace{-3pt}}),$ (2)
    $\displaystyle D({\hspace{-3pt}\begin{array}{c}\raisebox{-2pt}{\includegraphics[width=8mm]{figs/Lkink.eps}}\end{array}\hspace{-3pt}})$ $\displaystyle =$ $\displaystyle a^{-1}D({\hspace{-3pt}\begin{array}{c}\raisebox{1pt}{\includegraphics[width=8mm]{figs/Nkink.eps}}\end{array}\hspace{-3pt}}),$ (3)
    $\displaystyle D(\backoverslash) - D(\slashoverback)$ $\displaystyle =$ $\displaystyle z(D(\hsmoothing) - D(\smoothing)).$ (4)

    1. Compute $ D(\bigcirc^k)$ (where $ \bigcirc^k$ is the $ k$-component unlink).

      Hint. One instance of relation (4) relates the following four knots; three of them are the unknot with different framings:

      $\displaystyle {\hspace{-3pt}\begin{array}{c}\raisebox{0mm}{\includegraphics[width=3in]{figs/FourKnots.eps}}\end{array}\hspace{-3pt}} $

    2. Prove that the above conditions determine $ D$ on all knots and links.
    3. Set $ z=e^{x/4}-e^{-x/4}$ and $ a=\exp\left((N-1)\frac{x}{4}\right)$ and expand

      $\displaystyle D(K; z, a) = \sum_{m=0}^\infty D_m(K; N)x^m $

      (here $ K$ stands for an arbitrary knot or link). Prove that for any $ m$ the coefficient $ D_m$ is a type $ m$ invariant of links with values in polynomials in $ N$.
    4. Determine the weight system of $ D_m$ and show that it is the weight system arising from the Lie algebra $ so(N)$.

  3. Claim: The integral operator given by the kernel

    $\displaystyle G_{ij}(x,y) =
\frac{x^k-y^k}{\vert x-y\vert^3}

    is an inverse of the differential operator $ \star d$.

    Explain what this claim means and prove it. This done, show that if $ \gamma_{1,2}$ are disjoint space curves, then

    $\displaystyle \int dt_1dt_2 
G_{ij}(\gamma_1(t_1), \gamma_2(t_2))
= \int_{T^2}\Phi^\star\omega,

    where $ \Phi:T^2\to S^2$ is the ``direction of sight'' map $ \Phi(t_1,t_2) =
\frac{\gamma_1(t_1)-\gamma_2(t_2)}{\vert\gamma_1(t_1)-\gamma_2(t_2)\vert}$ and where $ \omega$ is the volume form of $ S^2$ normalized so that the total volume of $ S^2$ is $ 1$.

Good Luck!!

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Dror Bar-Natan 2003-12-19