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Geometric structures of 3-manifolds and various related structures

Research Project

Project/Area Number 17540077
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionNara Women's University

Principal Investigator

KOBAYASHI Tsuyoshi  Nara Women's University, Graduate School of Humanities and Sciences, Professor, 大学院人間文化研究科, 教授 (00186751)

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)
Project Period (FY) 2005 – 2006
Project Status Completed (Fiscal Year 2006)
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)
Keywordsknot / tunnel number / Seifert surface / automatic group / growth function / Heegaard splitting / 結び目、絡み目 / 橋指数 / Heegaard分解 / Seifert曲面 / plumbing
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.

Report

(3 results)
  • 2006 Annual Research Report   Final Research Report Summary
  • 2005 Annual Research Report

Research Products

(16 results)

All 2006 2005

All Journal Article (16 results)

  • [Journal Article] Heegaard genus of the connected sum of m-small knots2006

    • Author(s)
      Tsuyoshi Kobayashi
    • Journal Title

      Communications in Analysis and Geometry 14

      Pages: 1037-1077

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Annual Research Report 2006 Final Research Report Summary
  • [Journal Article] On the growth rate of tunnel number of knots2006

    • Author(s)
      Tsuyoshi Kobayashi
    • Journal Title

      J. reine angew. Math. 592

      Pages: 63-78

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Annual Research Report 2006 Final Research Report Summary
  • [Journal Article] Drawing Bers embeddings of the Teichmuller space of once-punctured tori2006

    • Author(s)
      Y.Komori
    • Journal Title

      Experimental Math. 15

      Pages: 50-60

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Annual Research Report 2006 Final Research Report Summary
  • [Journal Article] Computer experiments on the discreteness locus in projective structures2006

    • Author(s)
      Yasushi Yamashita
    • Journal Title

      Lond. Math. Soc. Lee. Notes 329

      Pages: 375-390

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Drawing Bers embeddings of the Teichmuller space of once-punctured tori2006

    • Author(s)
      Y.Komori, T.Sugawa, M.Wada, Y.Yamashita
    • Journal Title

      Experimental Math. 15

      Pages: 50-60

    • NAID

      110000164658

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Computer experiments on the discreteness locus in projective structures2006

    • Author(s)
      Y.Yamashita
    • Journal Title

      Lond. Math. Soc. Lec. Notes 329

      Pages: 375-390

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Heegaard genus of the connected sum of m-small knots2006

    • Author(s)
      T.Kobayashi, Y.Riek
    • Journal Title

      Communications in Analysis and Geometry 14

      Pages: 1037-1077

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] On the growth rate of tunnel number of knots2006

    • Author(s)
      T.Kobayashi, Y.Riek
    • Journal Title

      Journal fur die Reine and Ang. Math. 592

      Pages: 63-78

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Computer experiments on the discreteness locus in projective structures2006

    • Author(s)
      Yasushi Yamashita
    • Journal Title

      Lond. Math. Soc. Lec. Notes 329

      Pages: 375-390

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Computer experiments on the discreteness locus in projective structures2006

    • Author(s)
      Yamashita Yasushi
    • Journal Title

      Lond.Math.Soc.Lee.Notes 329

      Pages: 375-390

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Essential laminations and branched surfaces in the exteriors of links2005

    • Author(s)
      M.Brittenham
    • Journal Title

      Japanese J. Math. 31

      Pages: 25-96

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] A search method for a thin position of a link2005

    • Author(s)
      Tsuyoshi Kobayashi
    • Journal Title

      Algebr. Geom. Topol. 5

      Pages: 1027-1050

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Essential laminations and branched surfaces in the exteriors of links2005

    • Author(s)
      M.Brittenham, C.Hayashi, M.Hirasawa, T.Kobayashi, K.Shimokawa
    • Journal Title

      Japanese J. Math. 31

      Pages: 25-96

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] A search method for a thin position of a link2005

    • Author(s)
      T.Kobayashi, D.Heath
    • Journal Title

      Algebr. Geom. Topol. 5

      Pages: 1027-1050

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Essential laminations and branched surfaces in the exteriors of links2005

    • Author(s)
      Tsuyoshi Kobayashi
    • Journal Title

      Japanese J.Math. 31

      Pages: 25-96

    • Related Report
      2005 Annual Research Report
  • [Journal Article] A search method for a thin position of a link2005

    • Author(s)
      Tsuyoshi Kobayashi
    • Journal Title

      Algebr.Geom.Topol. 5

      Pages: 1027-1050

    • Related Report
      2005 Annual Research Report

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Published: 2005-03-31   Modified: 2016-04-21  

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