We investigate the question of when different surgeries on a knot can produce
identical manifolds. We show that given a knot in a homology sphere, unless
the knot is quite special, there is a bound on the number of slopes that
can produce a fixed manifold that depends only on this fixed manifold and
the homology sphere the knot is in. By finding a different bound on the
number of slopes, we show that non-null-homologous knots in certain homology
are determined by their complements. We also prove the surgery
characterisation of the unknot for null-homologous knots in
–spaces.
This leads to showing that all knots in some lens spaces are determined by
their complements. Finally, we establish that knots of genus greater than
in the Brieskorn
sphere
are also determined by their complements.
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