2006 Fiscal Year Final Research Report Summary
Exceptional Dehn surgery creating essential or Heegaard tori
| Project/Area Number |
16540071
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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 | Hiroshima University |
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Principal Investigator |
TERAGAITO Masakazu Hiroshima University, Graduate School of Education, Associate Professor, 大学院教育学研究科, 助教授 (80236984)
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| Co-Investigator(Kenkyū-buntansha) |
GODA Hiroshi Tokyo University of Agriculture and Technology, Institute of Symbiotic Science and Technology, Associate Professor, 大学院共生科学技術研究院, 助教授 (60266913)
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| Project Period (FY) |
2004 – 2006
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| Keywords | knot / 3-manifold / Dehn surgery |
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| Research Abstract |
In this research, we focused on exceptional Dehn surgery and filling for hyperbolic 3-manifolds which create essential tori or Heegaard tori. Precisely, we studied the following problems and obtained the results.
(1) Manifolds realizing the maximal distance between toroidal Dehn surgeries: First, we determined all knots that admit two toroidal Dehn surgeries at distance 5. Second, we showed that the distance between toroidal Dehn fillings on a large hyperbolic 3-manifold is at most 4. Finally, we proved that if a hyperbolic manifold admits a toroidal Dehn filling and an annular Dehn filling at distance 3, then the boundary of the manifold consists of at most two tori.
(2) Sequence of integral exceptional Dehn surgeries: Except Eudave-Munoz knots, it is conjectured that any exceptional Dehn surgery is integral. We studied all known examples of exceptional Dehn surgery, and found that integral exceptional Dehn surgeries form consecutive integers. By using the pentangle, we constructed hyperbolic knots which admit multiple exceptional Dehn surgeries.
(3) Non-integral Dehn surgeries creating closed non-orientable surfaces : We showed that any closed non-orientable surface of genus greater than two can be created by non-integral Dehn surgery on hyperbolic knots.
(4) Alexander polynomials of doubly primitive knots: We gave a formula of Alexander polynomial of doubly primitive knots. As a consequence, we showed that the Alexander polynomial of a doubly primitive knot has +1 and-1 alternatively as its coefficient.
(5) A Seifert fibered manifold with infinitely many knot-surgery descriptions: We gave the first example of a small Seifert fibered manifold that can be obtained from infinitely many hyperbolic knots by the same Dehn surgery. In particular, our knots have no symmetry, so that they cannot lie on a genus two Heegaard surface of the 3-sphere.
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