The Final Exam
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
- 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
is the -component
Hint. One instance of relation (4) relates
the following four knots; three of them are the unknot with different
- Prove that the above conditions determine on all knots and links.
(here stands for an arbitrary knot or link). Prove that for any
the coefficient is a type invariant of links with
values in polynomials in .
- Determine the weight system of and show that it is the weight
system arising from the Lie algebra .
- Claim: The integral operator given by the kernel
is an inverse of the differential operator .
Explain what this claim means and prove it. This done, show that if
are disjoint space curves, then
is the ``direction of sight'' map
where is the volume form of normalized so that the total
volume of is .
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