  For a knot K in S^3, let T(K) be the characteristic toric sub-orbifold of the
orbifold (S^3,K) as defined by Bonahon and Siebenmann. If K has unknotting
number one, we show that an unknotting arc for K can always be found which is
disjoint from T(K), unless either K is an EM-knot (of Eudave-Munoz) or (S^3,K)
contains an EM-tangle after cutting along T(K). As a consequence, we describe
exactly which large algebraic knots (ie algebraic in the sense of Conway and
containing an essential Conway sphere) have unknotting number one and give a
practical procedure for deciding this (as well as determining an unknotting
crossing). Among the knots up to 11 crossings in Conway's table which are
obviously large algebraic by virtue of their description in the Conway
notation, we determine which have unknotting number one. Combined with the work
of Ozsvath-Szabo, this determines the knots with 10 or fewer crossings that
have unknotting number one. We show that an alternating, large algebraic knot
with unknotting number one can always be unknotted in an alternating diagram.
  As part of the above work, we determine the hyperbolic knots in a solid torus
which admit a non-integral, toroidal Dehn surgery. Finally, we show that having
unknotting number one is invariant under mutation.
