For a knot
in , let
be the characteristic toric sub-orbifold of the orbifold
as defined by
Bonahon–Siebenmann. If
has unknotting number one, we show that an unknotting arc for
can always be found which
is disjoint from , unless
either is an EM–knot
(of Eudave-Muñoz) or
contains an EM–tangle after cutting along
. 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 Ozsváth–Szabó, 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.
Keywords
unknotting number 1, Conway spheres, algebraic knots,
mutation
