|Budget Amount *help
¥1,500,000 (Direct Cost : ¥1,500,000)
Fiscal Year 1999 : ¥500,000 (Direct Cost : ¥500,000)
Fiscal Year 1998 : ¥1,000,000 (Direct Cost : ¥1,000,000)
W. Thurston's Hyperbolic Surgery Theorem shows that all but a finite number of Dehn surgeries on a hyperbolic knot yield hyperbolic manifolds. It is thus important to study those exceptional surgeries yielding non-hyperbolic manifolds. By the join work with Kimihiko Motegi (Nihon Univ.) the author studied when surgey yields a Seifer fibered manifold or a manifold containing essential tours, typical examples of non-hyperbolic manifolds, and obtained the following results.
1. It had been already known that no Dehn surgery on a knot with period greater than 2 does not yield a Seifert fibered manifold. Regarding a knot with period 2, we showed that if such a knot yeilds a Seifert fibered manifold, then the quotient of the knot by its periodic map is a torus knot.
2. When does Dehn surgery on a hyperbolic, periodic knot yield a manifold containing an essential torus? We showed that a knot with period greater than 2 admits such a surgery if and only if its genus is 1, its period is 3, and the surgery coefficient is 0; if a knot with period 2 admits such a surgery, then the surgery coefficient is integer.
3. What Seifert fibered manifolds are obtined by Dehn surgery? We studied this problem in terms of indices of exceptinal fibers of Seifert fibered manifolds, and showed that : for arbitary integers p, q, r with some pair coprime, a surgery on some hyperbolic knot yields a Seifert fibered manifold with 3 exceptional fibers of indices p, q, r.