Dror Bar-Natan: Classes: 2003-04: Math 1350F - Knot Theory: | (51) |
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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!

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

- The ``Dubrovnik Polynomial'' (a variant of the ``Kauffman
Polynomial'' ) is an invariant of framed links valued in rational
functions in the variables and , satisfying the following
relations:

- Compute
(where
is the -component
unlink).
Hint. One instance of relation (4) relates the following four knots; three of them are the unknot with different framings:

- Prove that the above conditions determine on all knots and links.
- Set
and
and expand
- Determine the weight system of and show that it is the weight system arising from the Lie algebra .

- Compute
(where
is the -component
unlink).
- Claim: The integral operator given by the kernel
Explain what this claim means and prove it. This done, show that if are disjoint space curves, then

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Dror Bar-Natan 2003-12-19