| Co-Investigator(Kenkyū-buntansha) |
YAMASHITA Yasushi Nara Women's University, Faculty of Science, Associate Professor, 理学部, 助教授 (70239987)
KATAGIRI Minnyou Nara Women's University, Faculty of Science, Associate Professor, 理学部, 助教授 (60263422)
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| Budget Amount *help |
¥2,200,000 (Direct Cost: ¥2,200,000)
Fiscal Year 2006: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2005: ¥1,200,000 (Direct Cost: ¥1,200,000)
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| Research Abstract |
In this research project, we obtained the following results.
1. We defined a numerical invariant, called growth rate of tunnel numbers, of knots in 3-manifolds. For m-small knots, we obtained the following.
Suppose K is a m-small knot in. a 3-manifold M. Let g = g(X)-g(M), and b_i (i =1,..., g) be the bridge index of K with respect to genus g(X) - i Heegaard surface of M. Then the growth rate of K is given by min_i=_<1,..., n>{1-i/(b_i)}.
2. Heegaard splittings of exteriors of knots.
・ Let K_1, K_2 be knots in 3-manifolds, and T_1,T_2 tunnel systems of K_1, K_2 respectively. We gave a necessary and sufficient condition for the tunnel system t_1 ∪ T_2 of K_1#K_2 giving a stabilized Heegaard splitting.
・ For each natural number n, there exists a knot K such that the equality g(nK) = gt(K) holds, where nK denotes the connected sum of n copies of K. This implies the existence of counterexample to Morimoto's Conjecture concerning super additive phenomina of tunnel number of knots.
3. We showed that for any link L in the 3-sphere, there is a Seifert surface S for L such that S is obtained from a disk by successively plumbing flat annuli, where all of the attaching regions are contained in the disk.
4. We made research on Gersten's Problem : each automatic group is either (1) a finite group, (2) contains a free abelian group of rank 2. or (3) a word hyperbolic group.
We showed that for the n-starred automatic groups this assertion holds.
5. Growth function of 2-bridge link groups
We made computar experiments on the growth functions of 2-bridge link groups, and posed conjectures on the structure of the growth functions.
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