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三次元多様体の多重値モース理論の構築

Research Project

Project/Area Number 15740031
Research Category

Grant-in-Aid for Young Scientists (B)

Allocation TypeSingle-year Grants
Research Field Geometry
Research InstitutionTokyo University of Agriculture and Technology

Principal Investigator

合田 洋  東京農工大学, 大学院・共生科学技術研究部, 助教授 (60266913)

Project Period (FY) 2003 – 2005
Project Status Completed (Fiscal Year 2005)
Budget Amount *help
¥3,100,000 (Direct Cost: ¥3,100,000)
Fiscal Year 2005: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2004: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2003: ¥1,100,000 (Direct Cost: ¥1,100,000)
Keywords三次元多様体 / モース理論 / 結び目 / アレキサンダー多項式
Research Abstract

三次元Seiberg-Witten不変量と(通常の)ライデマイスタートーションΔ(t),ノビコフ複体に対するライデマイスタートーションτ(t)とレフセッツ・ゼータ関数ζ(t)との関係が知られている.三次元Seiberg-Witten不変量は各スピン構造をfixして値が決まるので,それらの‘平均'を<SW>___-と書くことにすると,
<SW>___-=Δ(t)=τ(t)・ζ(t)
となる.結び目の世界ではΔ(t)はアレキサンダー多項式というよく知られた結び目不変量である.昨年度はこのτ(t)とζ(t)の計算方法を確立した.具体的には,結び目補空間を縫い目付き多様体とみなしたときに適応されるヘガード分解のヘガードダイアグラムからτ(t)を導く方法を見出し,一方でζ(t)はこのヘガード分解に付随したflowのうち,閉軌道となるものをゼータ関数を使って数え上げるあるレフセッツ・ゼータ関数になることを突き止めた.
本年度は,まず一つの三次元多様体Mを固定したとき,このMに対するヘガード分解は一般のヘガード分解同様,stabilizationという操作で全て移りあうことを証明した.そして,このstabilizationの元で,τ(t),ζ(t)の変化について調べた.stabilizationによってτ(t)がτ'(t)に,ζ(t)がζ'(t)に変化したとすると:
τ'(t)=(τ(t))/(1+t^2),ζ'(t)=(1+t^2)ζ(t)
が成立する.
一方で,twist knotsや特別なpretzel knotsに対して,τ(t),ζ(t)の公式を得ることに成功した.この計算には各結び目のザイフェルト曲面が重要な役割をはたすので,pretzel knotsのザイフェルト曲面の分類も行った.

Report

(3 results)
  • 2005 Annual Research Report
  • 2004 Annual Research Report
  • 2003 Annual Research Report
  • Research Products

    (12 results)

All 2006 2005 2004 Other

All Journal Article (9 results) Publications (3 results)

  • [Journal Article] Circle valued Morse theory for knots and links2006

    • Author(s)
      Hiroshi Goda
    • Journal Title

      Clay Mathematics Proceedings (印刷中)

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Knot Floer Homology of (1,1)-knots2005

    • Author(s)
      Hiroshi Goda, Hiroshi Matsuda, Takayuki Morifuji
    • Journal Title

      Geometriae Dedicata 112

      Pages: 197-214

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Twisted Novikov homology and Circle-valued Morse theory for knots and links2005

    • Author(s)
      Hiroshi Goda, Andrei Pajitnov
    • Journal Title

      Osaka Journal of Mathematics 42

      Pages: 557-572

    • NAID

      120005986867

    • Related Report
      2005 Annual Research Report
  • [Journal Article] On hyperbolic 3-manifolds realizing the maximal distance between toroidal Dehn fillings2005

    • Author(s)
      Hiroshi Goda, Masakazu Teragaito
    • Journal Title

      Algebraic and Geometric Topology 5

      Pages: 463-507

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Reidemeister torsion, twisted Alexander polynomial and fibered knots2005

    • Author(s)
      Hiroshi Goda, Teruaki Kitano, Takayuki Morifuji
    • Journal Title

      Commentarii Mathematici Helvetici 80

      Pages: 51-61

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Knot Floer Homology of (1,1)-knots2005

    • Author(s)
      Hiroshi Goda, Hiroshi Matsuda, Takayuki Morifuji
    • Journal Title

      Geometriae Dedicata (印刷中)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Twisted Novikov homology and Circle-valued Morse theory for knots and links2005

    • Author(s)
      Hiroshi Goda, Andrei Pajitnov
    • Journal Title

      Osaka Journal of Mathematics 42(印刷中)

    • NAID

      120005986867

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Circle valued Morse theory for knots and links2005

    • Author(s)
      Hiroshi Goda
    • Journal Title

      Clay Mathematics Proceedings (印刷中)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] A criterion for satellite 1-genus 1-bridge knots2004

    • Author(s)
      Hiroshi Goda, Chuichiro Hayashi, Hyun-Jong Song
    • Journal Title

      Proceedings of the American Mathematical Society 132

      Pages: 3449-3456

    • Related Report
      2004 Annual Research Report
  • [Publications] Hiroshi Goda: "Some topics on Hyperbolic geometry and Heegaard splittings of 3-maritiolds"Interdisciplinary Information Sciences. 9. 9 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Hiroshi Goda: "Twisted alexander polynomial for SL(2,C)-representations and fibered knots"Comptes rendus mathematiques de I'Academie des Sciences. 25. 5 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Hiroshi Goda: "A criterion for satellite 1-genus 1-bridge Knots"Proceedings of the American Mathematical Society. (発表予定).

    • Related Report
      2003 Annual Research Report

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Published: 2003-04-01   Modified: 2016-04-21  

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