| Budget Amount *help |
¥4,010,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥510,000)
Fiscal Year 2007: ¥2,210,000 (Direct Cost: ¥1,700,000、Indirect Cost: ¥510,000)
Fiscal Year 2006: ¥1,800,000 (Direct Cost: ¥1,800,000)
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| Research Abstract |
For each multi-index I which consists of 1,...,m, the Milnor invariant μ(I) is defined. Let r(I) be the maximum number of times that any index appears in I. We investigate an equivalence relation, self Cn-equivalence, of links which is a generalized link-homotopy. Here, link-homotopy, which is defined by Milnor, is a well-known equivalence relation of links, and μ(I) is a link-homotopy invariant for any I with r(I)=1. We remark that self Cn-equivalence coincides with link-homotopy if n=1.
As a result of during the research of 2006, we have that μ(I) is a self Cn-equivalence invariant if(I)≦n.
It is known that, for strip links, Milnor invariants with r(I)=1 give the link-homotopy classification. This lets us think of a question : Do Milnor invariants with r(I)≦n give the self Cn-equivalence classification of string links?
During the research of 2007, we have a negative answer to the question above even for n=2. We also have that a link is self C_2-equivalent to the trivial link if and only if its Milnor invariants with r(I)≦2 vanish. So Milnor invariants with r(I)≦2 are strong enough to show that a link is self C_2-equivalent to the trivial link, although they are not enough to classify string links up to self C_2-equivalence.
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