We introduce the notions of Heegaard splittings and thin multiple Heegaard splittings of 1-submanifolds in compact orientable 3-manifolds, which are generalizations of those of bridge decompositions and thin positions. We show that either a thin multiple Heegaard splitting of 1-submanifold T is also a Heegaard splitting with minimal complexity or the exterior of T contains an essential surface with meridional boundary other than the boundary parallel annulus.

Dedicated to Professor Yukio Matsumoto for his 60th birthday.I would like to thank him for his encouragement, and letting me study anything I like when I was a student.Abstract. How many Reidemeister moves do we need for unknotting a given diagram of the trivial knot? Hass and Lagarias gave an upper bound. We give an upper bound for deforming a diagram of a split link to be disconnected.On the other hand, the absolute value of the writhe gives a lower bound of the number of Reidemeister I moves for unknotting. That of a complexity of knot diagram "cowrithe" works for Reidemeister II, III moves.We give an example of an infinite sequence of diagrams D n of the trivial knot with an O(n) number of crossings such that the author expects the number of Reidemeister moves needed for unknotting it to be O(n 2 ). However, writhe and cowrithe do not prove this. An upper bound for the number of Reidemeister moves for unlinkingA Reidemeister move is a local move of a link diagram as in Figure 1. Any such move does not change the link type. As Alexander and Briggs [1] and Reidemeister [7] showed that, for any pair of diagrams D 1 , D 2 which represent the same link type, there is a finite sequence of Reidemeister moves which deforms D 1 to D 2 .Let D be a diagram of the trivial knot. We consider sequences of Reidemeister moves which unknot D, i.e., deform D to have no crossing. Over all such sequences, we set ur(D) to be the minimal number of the moves in a sequence. Then let ur(n) denote the maximum ur(D) over all digrams of the trivial knot with n crossings. In [3], J. Hass and J. Lagarias gave an upper bound for ur(n), showing that ur(n) ≤ 2 cn , where c = 10 11 . (See also [2].)

We completely determine which Dehn surgeries on 2-bridge links yield reducible 3-manifolds. Further, we consider which surgery on one component of a 2-bridge link yields a torus knot, a cable knot and a satellite knot in this paper.

The cabling conjecture states that a non-trivial knot K in the 3-sphere is a cable knot or a torus knot if some Dehn surgery on K yields a reducible 3-manifold. We prove that symmetric knots satisfy this conjecture. (Gordon and Luecke also prove this independently ([GLu3]), by a method different from ours.)

We show that any Heegaard splitting of trivial arcs in a compression body is standard.