A slope p/q is a characterizing slope for a knot K in S-3 if the oriented homeomorphism type of p/q-surgery on K determines K uniquely. We show that for each torus knot its set of characterizing slopes contains all but finitely many non-integer slopes. This generalizes work of Ni and Zhang who established such a result for T-5,T-2. Along the way we show that if two knots K and K' in S-3 have homeomorphic p/q-surgeries, then for q >= 3 and p sufficiently large we can conclude that K and K' have the same genera and Alexander polynomials. This is achieved by consideration of the absolute grading on Heegaard Floer homology.
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