Chuichiro Hayashi

Only single twists on unknots can produce composite knots

Let K be a knot in the 3-sphere S3, and D a disc in S3 meeting K transversely more than once in the interior. For non-triviality we assume that IK n DI > 2 over all isotopy of K. Let Kn (C S3) be a knot obtained from K by cutting and n-twisting along the disc D (or equivalently, performing 1/n-Dehn surgery on OD). Then we prove the following: (1) If K is a trivial knot and Kn is a composite knot, then Inj < 1; (2) if K is a composite knot without locally knotted arc in S3 -,OD and Kn is also a composite knot, then Inj < 2. We exhibit some examples which demonstrate that both results are sharp. Independently Chaim Goodman-Strauss has obtained similar results in a quite different method.

Genus one 1-bridge positions for the trivial knot and cabled knots

Let ( V i , t i ) be a pair consisting of a solid torus V i and an unknotted arc t i properly imbedded in V i for i =1 and 2. Taking the union of these pairs we obtain a pair ( M , K ) of the 3-dimensional sphere or a lens space M and a knot K . We call such a knot K a genus one 1-bridge knot. In this paper we determine when a genus one 1-bridge knot is the trivial knot or a cabled knot.

DEHN SURGERY ON KNOTS IN SOLID TORI CREATING ESSENTIAL ANNULI

Let M be a 3-manifold obtained by performing a Dehn surgery on a knot in a solid torus. In the present paper we study when M contains a separating essential annulus. It is shown that M does not contain such an annulus in the majority of cases. As a corollary, we prove that symmetric knots in the 3-sphere which are not periodic knots of period 2 satisfy the cabling conjecture. This is an improvement of a result of Luft and Zhang. We have one more application to a problem on Dehn surgeries on knots producing a Seifert fibred manifold over the 2-sphere with exactly three exceptional fibres.

CONSTRUCTING LINKS BY PLUMBING FLAT ANNULI

We show that every oriented link is the boundary of a surface obtained from an embedded disk by plumbing a flat annulus a finite number of times.

MINIMAL UNKNOTTING SEQUENCES OF REIDEMEISTER MOVES CONTAINING UNMATCHED RII MOVES

Arnold introduced invariants

HEEGAARD SPLITTINGS OF THE TRIVIAL KNOT

J^+

Minimal sequences of Reidemeister moves on diagrams of torus knots

,

DEHN SURGERIES ON 2-BRIDGE LINKS WHICH YIELD REDUCIBLE 3-MANIFOLDS

J^-

Tangles and tubing operations

and

ON LINEAR n-COLORINGS FOR KNOTS

St