2006 Fiscal Year Final Research Report Summary
Research of relation among quantum invariant and number theoretic invariants and modular forms
| Project/Area Number |
17540067
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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 | Niigata University |
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Principal Investigator |
TAKATA Toshie Niigata University, Institute of Science and Technology, Associate Professor, 自然科学系, 助教授 (40253398)
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| Co-Investigator(Kenkyū-buntansha) |
AKIYAMA Shigeki Niigata University, Institute of Science and Technology, Associate Professor, 自然科学系, 助教授 (60212445)
HIKAMI Kazuhiro University of Tokyo, Graduate School of Science, Assistant, 大学院理学系研究科, 助手 (60262151)
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| Project Period (FY) |
2005 – 2006
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| Keywords | 3-manifold / knot / quantum invariant / modular form / theta function |
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| Research Abstract |
Applying to our formula of the colored Jones polynomial for 2-bridge knots Lawrence and Ron's way to compute SU(2)Witten--Reshetikhin--Turaev (WRT) invariant of homology 3-sphere obtained by surgery of a knot from the colored Jones polynomial of the knot, we conjectured some number theoretical properties of the Ohtsuki invariants. Later on, using some results about the colored Jones polynomial given by K.Habiro, we showed the conjecture. Furthermore, using the values of the Ohtsuki invariants for Seifert 3-manifolds given by Hikami, we identified the set of the LMO invariant up to degree 6 for integral homology 3-spheres, and characterized the set of the Ohtsuki invariants up to degree 6.
Lawrence and Zagier pointed out that WRT invariant of the Poincare homology sphere is related to the vector modular form with half-integral weight. Namely the WRT invariant coincides with a certain limit of the Eichler integral of the modular forms. As a generalization of this result, we have computed the WRT invariant for the spherical Seifert manifold with three exceptional fibers, and we have found that it coincides with a limiting value of Eicher integral of vector modular form with half-integral weight. By use of the modular property, we have obtained an exact asymptotic expansion of the WRT invariant, and have given an interpretation of topological invariants such as the Ohtsuki series and the Chern--Simons invariant from the viewpoint of modular forms. Furthermore, we have pointed out that the vector modular form is related to the fundamental group of manifolds.
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Research Products
(12 results)
