  This thesis is concerned with the question of when the double branched cover
of an alternating knot can arise by Dehn surgery on a knot in $S^3$. We
approach this problem using a surgery obstruction, first developed by Greene,
which combines Donaldson's Diagonalization Theorem with the $d$-invariants of
Ozsv{\'a}th and Szab{\'o}'s Heegaard Floer homology. This obstruction shows
that if the double branched cover of an alternating knot or link $L$ arises by
surgery on $S^3$, then for any alternating diagram the lattice associated to
the Goeritz matrix takes the form of a changemaker lattice. By analyzing the
structure of changemaker lattices, we show that the double branched cover of
$L$ arises by non-integer surgery on $S^3$ if and only if $L$ has an
alternating diagram which can be obtained by rational tangle replacement on an
almost-alternating diagram of the unknot. When one considers half-integer
surgery the resulting tangle replacement is simply a crossing change. This
allows us to show that an alternating knot has unknotting number one if and
only if it has an unknotting crossing in every alternating diagram.
  These techniques also produce several other interesting results: they have
applications to characterizing slopes of torus knots; they produce a new proof
for a theorem of Tsukamoto on the structure of almost-alternating diagrams of
the unknot; and they provide several bounds on surgeries producing the double
branched covers of alternating knots which are direct generalizations of
results previously known for lens space surgeries.
  The thesis begins with an exposition of the changemaker surgery obstruction,
giving an amalgamation of results due to Gibbons, Greene and the author. It
then gives background material on alternating knots and changemaker lattices.
The latter part of the thesis is then taken up with the applications of this
theory.

Geometric Topology