We determine all non-simple links which admit an unknotting tunnel,
i.e. links which contain an essential annulus or torus in its exterior
and have tunnel number one.
A motion of a link consists of an isotopy of the link through its ambient
space that ultimately returns the link to itself. Using the result on motions
of a trivial link, we can define an invariant of ribbon presentaions of
knots.
We determine $\Delta$-unknotting for torus knots, positive knots, and
positive closed 3-braid.
We define $b/a$-unknotting operation as a generalized unknotting operation.
And we determine all two-bridge knots with $b/a$-unknotting number one.
For an ordinary unknotting operation, Kanenobu and Murakami determined
all unknotting number one two-bridge knots. And P. Kohn determined all
unknotting number one two-bridge links.
We show that for many periodic knots its $\Delta$-unknotiing number
is greater than one.
We define the Gordian complex of the space
of oriented knot types, and announce the following: Let $K_0$ and $K_1$
be
any pair of knot types such that the Gordian distance $d_G(K_0;K_1)
= 1$.
Then there exists a family of knot types $\{K_0, K_1, K_2, \ldots,
K_n\}$ such that
$d_G(K_i;K_j ) = 1$ for any $ i < j$.
pdf file by M.Hirasawa.
Let $W$ be a genus two handlebody, $D$ is essential, if $D$ is properly embedded disk in $W$ and not $\partial$-pararell in $W$. Cutting $W$ along $D$, then we get a solid torus or two solid tori. In case of a solid torus, its core is a tunnel number one knot, in case of two solid tori, its core is a tunnel number one links. From this view, we will characteristic some tunnel number one knot.
Proceedings of International Conference on Topology in Matsue,Topology and its Applications 146-147 (2005) 189-194
