Study of perturbative quantum invariant of rational homology 3-spheres

• Project/Area Number: 22540073
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Geometry
• Research Institution: Kyushu University
• Principal Investigator: TAKATA Toshie 九州大学, 数理学研究院, 准教授 (40253398)
• Project Period (FY): 2010 – 2012
• Project Status: Completed (Fiscal Year 2012)
• Budget Amount: ¥3,250,000 (Direct Cost: ¥2,500,000、Indirect Cost: ¥750,000); Fiscal Year 2012: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2011: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2010: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
• Keywords: 低次元トポロジー / 量子不変量 / ３次元多様体 / free energy / 3次元多様体 / LMO不変量 / weight system

Research Abstract

For a rational homology 3-sphre, the SU(N) free energy is a power series in 2 variables and an topological invariant of M. The SU(N) free energy can be obtained by applying the SU(N) weight system to the LMO invariant of M. By using a combinatorial way to compute the SU(N) weight system, we obtained an formula for any genus of the SU(N) free energy for Seifert homology 3-spheres. It can be written in terms of the generalized Catalan numbers. Especially, the genus 0 part of the free energy can be written in terms of the generalized Catalan numbers for genus 0 where parameters are even numbers. We gave a formula of these generalized Catalan numbers.

Research Products

• [Journal Article] On the SO(N) and Sp(N) free energy of a rational homology 3-sphere (2011)
  • Author(s): 高田敏恵
  • Journal Title: International Journal of Mathematics Volume: 22 Pages: 465-482
  • Peer Reviewed
• [Journal Article] On the SO(N) and Sp(N) free energy of a rational homology 3-sphere (2011)
  • Author(s): 高田敏恵
  • Journal Title: International Joounal of Mathematics Volume: 22 Issue: 04 Pages: 465-482
  • DOI: 10.1142/s0129167x11006866
  • Peer Reviewed
• [Presentation] An algebraic formula of the Witten-Reshetikihin-Turaev invariant for 3-manifolds (2012)
  • Author(s): 高田敏恵
  • Organizer: The 8th East Asian School of Knots and Related Topics
  • Place of Presentation: KAIST(韓国)
  • Year and Date: 2012-01-09
• [Presentation] An algebraic formula of the Witten-Reshetikihin-Turaev invariant for 3-manifods (2012)
  • Author(s): 高田敏恵
  • Organizer: The 8the East Asian School of Knots and Related Topics
  • Place of Presentation: KAIST(韓国)
  • Year and Date: 2012-01-09
• [Presentation] Ribbon Hopf algebras と3次元多様体の量子不変量 (2011)
  • Author(s): 高田敏恵
  • Organizer: 結び目の量子不変量とそのカテゴリー化
  • Place of Presentation: 早稲田大学
  • Year and Date: 2011-08-24
• [Presentation] Ribbon Hopf algebrasと3次元多様体の量子不変量 (2011)
  • Author(s): 高田敏恵
  • Organizer: 結び目の量子不変量とそのカテゴリー化
  • Place of Presentation: 早稲田大学(東京)
  • Year and Date: 2011-08-24
• [Presentation] 3 次元多様体の LMO 不変量と摂動的不変量について (2010)
  • Author(s): 高田敏恵
  • Organizer: 日本数学会 2010 年度秋季総合分科会トポロジー分科会特別講演
  • Place of Presentation: 名古屋大学
  • Year and Date: 2010-09-22
• [Presentation] 3次元多様体のLMO不変量と摂動的不変量について (2010)
  • Author(s): 高田敏恵
  • Organizer: 日本数学会2010年度秋季総合分科会トポロジー分科会
  • Place of Presentation: 名古屋大学
  • Year and Date: 2010-09-22
• [Presentation] A relation between the HOMFLY and Kauffman polynomials for 2-component torus links (2010)
  • Author(s): 高田敏恵
  • Organizer: 広島大学トポロジー・幾何セミナー
  • Place of Presentation: 広島大学
  • Year and Date: 2010-07-13
• [Presentation] On the SO(N) and Sp(N) free energy of a rational homology 3-sphere (2010)
  • Author(s): 高田敏恵
  • Organizer: 量子群と量子トポロジー
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2010-04-20