# 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'' (a variant of the Kauffman Polynomial'' ) is an invariant of framed links valued in rational functions in the variables and , satisfying the following relations:
 (1) (2) (3) (4)

1. 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:

2. Prove that the above conditions determine on all knots and links.
3. Set and and expand

(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 .
4. Determine the weight system of and show that it is the weight system arising from the Lie algebra .

3. 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

where is the direction of sight'' map and where is the volume form of normalized so that the total volume of is .

Good Luck!!

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Dror Bar-Natan 2003-12-19