A study of the fiberedness and the genus using character varieties of knot groups

• Project/Area Number: 23540076
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Keio University (2012-2014); Tokyo University of Agriculture and Technology (2011)
• Principal Investigator: MORIFUJI Takayuki 慶應義塾大学, 経済学部, 教授 (90334466)
• Project Period (FY): 2011-04-28 – 2015-03-31
• Project Status: Completed (Fiscal Year 2014)
• Budget Amount: ¥4,030,000 (Direct Cost: ¥3,100,000、Indirect Cost: ¥930,000); Fiscal Year 2013: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2012: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2011: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
• Keywords: 結び目群 / 指標代数多様体 / ねじれアレキサンダー不変量

Outline of Final Research Achievements

The purpose of this research was to construct a framework for detecting geometric properties of knots by some kind of finiteness using the character variety of the knot group and the twisted Alexander invariant. The results are as follows.
(1) We showed that the fiberedness and the genus of certain wide classes of knots (which include 2-bridge knots) are determined by information of the twisted Alexander invariant on a one dimensional irreducible component of the character variety.
(2) We gave an affirmative answer to a conjecture of Dunfield, Friedl and Jackson about the fiberedness and the genus of hyperbolic knots for an infinite family of 2-bridge knots.

Research Products

• [Journal Article] Twisted Alexander polynomials of 2-bridge knots for parabolic representations (2014)
  • Author(s): Takayuki Morifuji, Anh T. Tran
  • Journal Title: Pacific Journal of Mathematics Volume: 269 Issue: 2 Pages: 433-451
  • DOI: 10.2140/pjm.2014.269.433
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Reducibility of automorphisms of Hurwitz surfaces and the eta-invariant (2014)
  • Author(s): Takayuki Morifuji
  • Journal Title: International Journal of Mathematics Volume: 25 Issue: 13 Pages: 14501191-8
  • DOI: 10.1142/s0129167x14501195
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Twisted Alexander polynomials on curves in character varieties of knot groups (2013)
  • Author(s): Taehee Kim, Takahiro Kitayama, Takayuki Morifuji
  • Journal Title: International Journal of Mathematics Volume: 24 Pages: 13500221-16
  • Peer Reviewed
• [Journal Article] A note on degrees of twisted Alexander polynomials II (2013)
  • Author(s): Takayuki Morifuji, Masaaki Suzuki
  • Journal Title: Bulletin of the Korean Mathematical Society Volume: 50 Pages: 929-934
  • Peer Reviewed
• [Journal Article] On the second coefficient of the twisted Alexander polynomial (2013)
  • Author(s): Takayuki Morifuji
  • Journal Title: RIMS Kokyuroku Volume: 1836 Pages: 104-109
• [Journal Article] Twisted Alexander polynomials and character varieties of 2-bridge knot groups (2012)
  • Author(s): Taehee Kim, Takayuki Morifuji
  • Journal Title: International Journal of Mathematics Volume: 23 Pages: 12500221-24
  • Peer Reviewed
• [Journal Article] Twisted Alexander polynomials for irreducible SL(2,C)-representations of torus knots (2012)
  • Author(s): Teruaki Kitano, Takayuki Morifuji
  • Journal Title: Annali della Scuola Normale Superiore di Pisa Classe di Scienze (5) Volume: 11 Pages: 395-406
  • Peer Reviewed
• [Journal Article] On a conjecture of Dunfield, Friedl and Jackson (2012)
  • Author(s): Takayuki Morifuji
  • Journal Title: Comptes Rendus de l'Academie des Sciences de Paris, Ser I Volume: 350 Pages: 921-924
  • Peer Reviewed
• [Journal Article] Twisted Alexander polynomials and character varieties of 2-bridge knots (2011)
  • Author(s): Taehee Kim, Takayuki Morifuji
  • Journal Title: RIMS Kokyuroku Volume: 1747 Pages: 77-80
• [Journal Article] A note on degrees of twisted Alexander polynomials (2011)
  • Author(s): Takayuki Morifuji
  • Journal Title: RIMS Kokyuroku Volume: 1747 Pages: 121-125
• [Presentation] On a conjecture of Dunfield-Friedl-Jackson for the hyperbolic torsion polynomial (2014)
  • Author(s): Takayuki Morifuji
  • Organizer: Topology Seminar
  • Place of Presentation: 韓国・建国大学（ソウル）
  • Year and Date: 2014-11-06
  • Invited
• [Presentation] 二橋結び目のパラボリック表現とねじれAlexander 多項式 (2014)
  • Author(s): 森藤 孝之
  • Organizer: 日本数学会2014 年度秋季総合分科会トポロジー分科会
  • Place of Presentation: 広島大学（広島）
  • Year and Date: 2014-09-28
• [Presentation] 基本群の表現空間とねじれAlexander不変量 (2012)
  • Author(s): 森藤孝之
  • Organizer: 上教大トポロジーセミナー(招待講演)
  • Place of Presentation: 上越教育大学
• [Presentation] Representation spaces and twisted Alexander polynomials of torus knots (2012)
  • Author(s): Takayuki Morifuji
  • Organizer: Le Teich, Laboratoire d'Analyse, Topologie, Probabilites(招待講演)
  • Place of Presentation: Aix-Marseille University
• [Presentation] 結び目群の表現とねじれAlexander多項式について (2011)
  • Author(s): 森藤孝之
  • Organizer: 東京工業大学トポロジーセミナー(招待講演)
  • Place of Presentation: 東京工業大学
• [Presentation] Representation spaces of torus knots and twisted Alexander polynomials (2011)
  • Author(s): Takayuki Morifuji
  • Organizer: GCOE Workshop ``Circle valued Morse theory and Alexander invariants''(招待講演)
  • Place of Presentation: 東京大学
• [Presentation] On twisted Alexander polynomials of torus knots for SL(2,C)-representations (2011)
  • Author(s): Takayuki Morifuji
  • Organizer: Topology Seminar(招待講演)
  • Place of Presentation: 北京大学
• [Presentation] 多項式不変量のカテゴリー化
  • Author(s): 森藤孝之
  • Organizer: 第60回 Encounter with Mathematics
  • Place of Presentation: 中央大学（東京都文京区）
  • Invited
• [Presentation] A formula for the twisted Alexander polynomial of torus knots
  • Author(s): Takayuki Morifuji
  • Organizer: 微分幾何・トポロジーセミナー
  • Place of Presentation: 慶應義塾大学
  • Invited
• [Presentation] Representation spaces of knot groups and twisted Alexander invariants
  • Author(s): Takayuki Morifuji
  • Organizer: The Conference of Group Actions and Applications in Geometry, Topology and Analysis
  • Place of Presentation: The Center of Engineering Mathematics, Kunming University of Science and Technology, China
  • Invited
• [Presentation] The hyperbolic torsion polynomial of twist knots
  • Author(s): Takayuki Morifuji
  • Organizer: The 9th East Asian School of Knots and Related Topics
  • Place of Presentation: University of Tokyo
• [Presentation] On twisted Alexander polynomials on curves
  • Author(s): Takayuki Morifuji
  • Organizer: トポロジーセミナー
  • Place of Presentation: 群馬大学
  • Invited
• [Presentation] The hyperbolic torsion polynomial and a conjecture of Dunfield-Friedl-Jackson
  • Author(s): Takayuki Morifuji
  • Organizer: Seminari di Geometria
  • Place of Presentation: Universita di Pisa, Italy
  • Invited