1998 Fiscal Year Final Research Report Summary
Perturbative expansion of quantum invariants
| Project/Area Number |
09640118
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | KYUSHU UNIVERSITY |
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Principal Investigator |
YOKOTA Yoshiyuki Kyushu Univ., Graduate School of Mathematics, Assistant Professor, 大学院・数理学研究科, 講師 (40240197)
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| Project Period (FY) |
1997 – 1998
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| Keywords | quantum invariant / incompressible surface / Heegaard splitting / rheta_m-curve |
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| Research Abstract |
The quantum invariants of 3-manifolds, which are defined for compact Lie groups, was first predicted by Witten, and rigorously established by Reshtikhin and Turaev. Although the invariants are complex-valued by definition, it has been believed to be algebraic integers. This is confirmed for Lie group SU(2) by Murakami, _from which Ohtsuki extracted an infinite series of 3-manifold invariants by so-called "algebraic perturbation". The purpose of this research was to construct such perturbative invariants of 3-manifolds for the other Lie groups, but this subject is independently acheived by Le. Thus, I proceed to the geometric application of the invariants, and obtain the following results.
1.Criteria for the incompressibility of surfaces in 3-manifolds
Although incompressible surfaces play important roles in 3-manifold theory, it has been difficult to detect the incom- pressibility of surfaces in manifolds. In this research, some criteria for such incompressibility of non-separating surfaces in manifolds are established in terms of representation matrices of mapping class groups derived from quantum invariants.
2.New invariants for Heegaard splittings of 3-manifolds
It is well-known that two Heegaard splittings of a 3-manifold are stably equivalent, but known invariants of Heegaard splittings behaves trivially under such equivalence. In this research some invariants are defined in terms of the elementary divisors of representation matrices of mapping class groups which behave nontrivially under stably equivalence.
3.Polynomial invariants for thetam-curves in 3-space
New polynomial invariants for thetam-curves in 3-space are introduced. The invariants are definitely computable, and can detect the chirality of graphs in which stereo-chemists are interested.
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