Dror Bar-Natan: Classes: 2003-04: Math 1350F - Knot Theory:

(51)

Next: Class Home Previous: Class Notes for Thursday December 4, 2003

The Final Exam

This document in PDF: Final.pdf

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:
   

   $$D(\bigcirc) = 1,$$ (1)

   $$D({\hspace{-3pt}\begin{array}{c}\raisebox{-2pt}{\includegraphics[width=8mm]{figs/Rkink.eps}}\end{array}\hspace{-3pt}}) = aD({\hspace{-3pt}\begin{array}{c}\raisebox{1pt}{\includegraphics[width=8mm]{figs/Nkink.eps}}\end{array}\hspace{-3pt}}),$$ (2)

   $$D({\hspace{-3pt}\begin{array}{c}\raisebox{-2pt}{\includegraphics[width=8mm]{figs/Lkink.eps}}\end{array}\hspace{-3pt}}) = a^{-1}D({\hspace{-3pt}\begin{array}{c}\raisebox{1pt}{\includegraphics[width=8mm]{figs/Nkink.eps}}\end{array}\hspace{-3pt}}),$$ (3)

   $$D(\backoverslash) - D(\slashoverback) = z(D(\hsmoothing) - D(\smoothing)).$$ (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:

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

   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
      $$D(K; z, a) = \sum_{m=0}^\infty D_m(K; N)x^m$$
      (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
   $$G_{ij}(x,y) = \frac{\epsilon_{ijk}}{4\pi} \frac{x^k-y^k}{\vert x-y\vert^3}$$
   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

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

The generation of this document was assisted by L^ATEX2HTML.