We show that if
–surgery
on a nontrivial knot
yields the branched double cover of an alternating knot, then
. This
generalises a bound for lens space surgeries first established by Rasmussen. We also show
that all surgery coefficients yielding the double branched covers of alternating knots
must be contained in an interval of width two and this full range can be realised only
if the knot is a cable knot. The work of Greene and Gibbons shows that if
bounds a sharp
–manifold
, then the
intersection form of
takes the form of a changemaker lattice. We extend this to show
that the intersection form is determined uniquely by the knot
, the slope
and the Betti
number
.
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