| Budget Amount *help |
¥1,700,000 (Direct Cost: ¥1,700,000)
Fiscal Year 2003: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2002: ¥900,000 (Direct Cost: ¥900,000)
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| Research Abstract |
In this project, we focused on toroidal surgery among exceptional Dehn surgery on hyperbolic 3-manifolds, and obtained several results. It is well known that a toroidal surgery of a hyperbolic knot in the 3-sphere corresponds to either an integer or a half-integer. First, we determined the simplest hyperbolic knot with non-integral toroidal surgery from a view of bridge index. That is, we showed that the (-2,3,7)-pretzel knot is the only 3-bridge knot with non integral toroidal surgery. Moreover, this knot is the only one with such surgery among pretzel knots. Second, we proposed a conjecture that any integral toroidal slope for a hyperbolic knot is bounded by the genus of the knot multiplied by four. In the first year, we solved this conjecture affirmatively for two important classes of knots, genus one knots and alternating knots. In the second year, we tried to prove the full conjecture, but we could not do this. But we have almost done. According to the minimal intersection number between an essential torus and the core of the attached solid torus, we got the expected upper bound, unless that number is not equal to four. Also, a toroidal surgery open creates a Klein bottle simultaneously. We got the best possible upper bound for Klein bottle surgery on all knots by using genera of knots. By using of this argument, we determined the crosscap numbers of torus knots and 2-bridge knots
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