Representation theoretical analysis of the knot contact homology

• Project/Area Number: 26800046
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Geometry
• Research Institution: Meijo University
• Principal Investigator: Fumikazu Nagasato 名城大学, 理工学部, 准教授 (30513634)
• Project Period (FY): 2014-04-01 – 2018-03-31
• Project Status: Completed (Fiscal Year 2017)
• Budget Amount: ¥2,600,000 (Direct Cost: ¥2,000,000、Indirect Cost: ¥600,000); Fiscal Year 2017: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2016: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2015: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2014: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
• Keywords: 結び目 / 基本群 / トレースフリー表現 / 指標多様体 / コンタクトホモロジー / ２重分岐被覆 / 幽霊指標 / 結び目群 / ノットコンタクトホモロジー / knot contact homology / character variety / ghost character / minimal element / トレースフリー断面 / 半順序

Outline of Final Research Achievements

This project focuses on a representation theoretical mechanism relating the abelian knot contact homology and the character ring of the 2-fold branched cover of a 3-sphere branched along a knot. This relationship was conjectured by L. Ng, and it has been established by him for 2-bridge knots. In this project, we first introduced a new notion called a ghost character of a knot, which gives an obstruction for the conjecture to be true. Using it, in a joint work with Shinnosuke Suzuki, who was a graduate student at Meijo University, we found a counterexample of the conjecture for some torus knots. We are still researching this relationship to reveal the representation theoretical mechanism of the knot contact homology completely.

Research Products

• [Int'l Joint Research] The University of Texas at Dallas(米国)
• [Int'l Joint Research] The University of Texas at Dallas(米国)
• [Journal Article] On minimality of two-bridge knots (2017)
  • Author(s): Fumikazu Nagasato, Masaaki Suzuki and Anh T. Tran
  • Journal Title: International Journal of Mathematics Volume: 28 Issue: 03 Pages: 1750020-1750020
  • DOI: 10.1142/s0129167x17500203
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] SOME FAMILIES OF MINIMAL ELEMENTS FOR A PARTIAL ORDERING ON PRIME KNOTS (2016)
  • Author(s): FUMIKAZU NAGASATO AND ANH T. TRAN
  • Journal Title: Osaka Journal of Mathematics Volume: 53
  • NAID: 120005986264
  • Peer Reviewed / Open Access / Int'l Joint Research / Acknowledgement Compliant
• [Presentation] Π1(∑2K)固有のSL(2,C)-表現とNg予想の反例について (2017)
  • Author(s): 長郷 文和，鈴木 心之助
  • Organizer: 第8回東名古屋数学若手研究集会
• [Presentation] Trace-free characters and abelian knot contact homology (On Ng's conjecture from ghost characters of a knot) (2017)
  • Author(s): Fumikazu Nagasato and Shinnosuke Suzuki
  • Organizer: the 2nd Pan-Pacific International Conference on Topology and Applications
  • Int'l Joint Research / Invited
• [Presentation] Ghost characters, character varieties and abelian knot contact homology (2017)
  • Author(s): Fumikazu Nagasato
  • Organizer: The 12th East Asian School of Knots and Related Topics
  • Place of Presentation: 東京大学，東京
  • Int'l Joint Research
• [Presentation] On a special SL(2,C)-representation of \pi_1(\Sigma_2K) (2017)
  • Author(s): Shinnosuke Suzuki
  • Organizer: The 12th East Asian School of Knots and Related Topics
  • Place of Presentation: 東京大学，東京
  • Int'l Joint Research
• [Presentation] On ghost characters of knot groups (2016)
  • Author(s): 長郷 文和
  • Organizer: 2016琉球結び目セミナー
  • Place of Presentation: 那覇市伝統工芸館
  • Year and Date: 2016-02-20
• [Presentation] Ghost characters, character varieties and abelian knot contact homology (2016)
  • Author(s): Fumikazu Nagasato
  • Organizer: KOOK-TAPU Workshop of Knots in Tsushima Island
  • Place of Presentation: 対馬市交流センター (TIARA)，長崎県対馬市
  • Int'l Joint Research
• [Presentation] On the SL(2,C)-representations of the fundamental group of the 2-fold branched covering of 3-sphere branched along a knot (2016)
  • Author(s): Shinnosuke Suzuki
  • Organizer: KOOK-TAPU Workshop of Knots in Tsushima Island
  • Place of Presentation: 対馬市交流センター (TIARA)，長崎県対馬市
  • Int'l Joint Research
• [Presentation] 幽霊指標，指標多様体と可換ノットコンタクトホモロジー (2016)
  • Author(s): Fumikazu Nagasato
  • Organizer: トポロジーとコンピュータ 2016
  • Place of Presentation: カレッジプラザ，秋田
• [Presentation] 結び目に沿って分岐する3次元球面の2重分岐被覆の基本群のSL(2,C)-表現について (2016)
  • Author(s): Shinnosuke Suzuki
  • Organizer: トポロジーとコンピュータ 2016
  • Place of Presentation: カレッジプラザ，秋田
• [Presentation] Ghost characters, character varieties and abelian knot contact homology (2016)
  • Author(s): Fumikazu Nagasato
  • Organizer: Fundamental Groups, Representations and Geometric Structures in 3-Manifold Topology
  • Place of Presentation: 広島大学，広島
  • Int'l Joint Research / Invited
• [Presentation] \pi_1(\Sigma_2K)固有のSL(2,C)-表現とNg予想の反例について (2016)
  • Author(s): Shinnosuke Suzuki
  • Organizer: 結び目の数学IX
  • Place of Presentation: 日本大学，東京
• [Presentation] On some families of minimal elements for the partial ordering on prime knots (2015)
  • Author(s): Fumikazu Nagasato
  • Organizer: The Tenth East Asian School of Knots and Related Topics
  • Place of Presentation: East China Normal University
  • Year and Date: 2015-01-26 – 2015-01-29
  • Invited
• [Presentation] On some families of minimal elements for the partial ordering on prime knots (2014)
  • Author(s): 長郷 文和
  • Organizer: 東北結び目セミナー 2014
  • Place of Presentation: カレッジプラザ
  • Year and Date: 2014-10-17 – 2014-10-19
• [Presentation] On "SL(2,C)-ghost characters'' of knots (2014)
  • Author(s): 長郷 文和
  • Organizer: 2014 琉球結び目セミナー
  • Place of Presentation: 那覇市伝統工芸館
  • Year and Date: 2014-06-21 – 2014-06-22