The existences of epimorphisms between knot groups and their geometric interpretations

• Project/Area Number: 16K05159
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Meiji University
• Principal Investigator: Suzuki Masaaki 明治大学, 総合数理学部, 専任教授 (70431616)
• Project Period (FY): 2016-04-01 – 2023-03-31
• Project Status: Completed (Fiscal Year 2022)
• Budget Amount: ¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000); Fiscal Year 2019: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2018: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2017: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2016: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
• Keywords: 結び目群 / 全射準同型 / 全射凖同型 / 低次元トポロジー

Outline of Final Research Achievements

A knot is an embedded circle. The knot group of a knot is a fundamental group of the exterior of a knot. The fundamental group is the set of loops where two loops are considered as the same element if they are transformed continuously. The knot group characterizes a knot. However, it is very complicated. Then in this research, we considered whether there exists an epimorphism between knot groups and their geometric interpretations.

Academic Significance and Societal Importance of the Research Achievements

結び目群は結び目をとてもよく特徴づけているが、一般に群を調べることは容易ではないことが多い。そのため、その間の全射準同型の存在という関係を調べることは意味がある。このようにそのものが単独では複雑で扱いにくいものでも、それらの関係を調べるということで研究を進めるという方法は様々な場面で応用できるものと考えられ、その点でも意義があると考える。

Research Products

• [Int'l Joint Research] The University of Texas at Dallas(米国)
• [Int'l Joint Research] University of Texas at Dallas(米国)
• [Int'l Joint Research] POSTECH(韓国)
• [Int'l Joint Research] University of Texas at Dallas(米国)
• [Int'l Joint Research] POSTECH(韓国)
• [Journal Article] Abelian quotients of the Y-filtration on the homology cylinders via the LMO functor (2022)
  • Author(s): Y. Nozaki, M. Sato, and M. Suzuki
  • Journal Title: Geometry and Topology Volume: 26 Pages: 221-282
  • Peer Reviewed
• [Journal Article] On the kernel of the surgery map restricted to the 1-loop part (2022)
  • Author(s): Y. Nozaki, M. Sato, and M. Suzuki
  • Journal Title: Journal of Topology Volume: 15 Pages: 587-619
  • Peer Reviewed
• [Journal Article] On a theorem of Friedl and Vidussi (2022)
  • Author(s): T. Morifuji and M. Suzuki
  • Journal Title: International Journal of Mathematics Volume: 33 Pages: 14-14
  • Peer Reviewed
• [Journal Article] Generating functions on epimorphisms between $2$-bridge knot groups (2021)
  • Author(s): Masaaki Suzuki
  • Journal Title: Journal of Knot Theory and Its Ramifications Volume: 30 Pages: 24-24
  • Peer Reviewed
• [Journal Article] Torelli group, Johnson kernel and invariants of homology spheres (2020)
  • Author(s): S. Morita, T. Sakasai, M. Suzuki
  • Journal Title: Quantum Topology Volume: 11 Pages: 379-410
  • Peer Reviewed
• [Journal Article] Alternating knots with Alexander polynomials having unexpected zeros (2019)
  • Author(s): M. Hirasawa, K. Ishikawa, M. Suzuki
  • Journal Title: Topology and its Applications Volume: 253 Pages: 48-56
  • Peer Reviewed
• [Journal Article] Genera of two-bridge knots and epimorphisms of their knot groups (2018)
  • Author(s): Anh, Tran and Masaaki Suzuki
  • Journal Title: Topology and its Applications Volume: 242 Pages: 66-72
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] An abelian quotient of the symplectic derivation Lie algebra of the free Lie algebra (2018)
  • Author(s): Shigeyuki Morita, Takuya Sakasai, and Masaaki Suzuki
  • Journal Title: Experimental Mathematics Volume: 27 Pages: 302-315
  • Peer Reviewed
• [Journal Article] Symmetry of symplectic derivation Lie algebra of free Lie algebras (2017)
  • Author(s): S. Morita, T. Sakasai, and M. Suzuki
  • Journal Title: RIMS Kokyuroku Bessatsu, Geometry and Analysis of Discrete Groups and Hyperbolic Spaces Volume: B66 Pages: 185-194
  • Peer Reviewed
• [Journal Article] Epimorphisms between 2-bridge knot groups and their crossing numbers (2017)
  • Author(s): Masaaki Suzuki
  • Journal Title: Algebraic and Geometric Topology Volume: 17 Pages: 2413-2428
  • Peer Reviewed
• [Journal Article] An abelian quotient of the symplectic derivation Lie algebra of the free Lie algebra (2017)
  • Author(s): Shigeyuki Morta, Takuya Sakasai, and Masaaki Suzuki
  • Journal Title: Experimental Mathematics Volume: 印刷中
  • Peer Reviewed
• [Journal Article] Symmetry of symplectic derivation Lie algebras of free Lie algebras (2017)
  • Author(s): Shigeyuki Morta, Takuya Sakasai, and Masaaki Suzuki
  • Journal Title: RIMS Kokyuroku Bessatsu Volume: 印刷中
  • Peer Reviewed
• [Journal Article] On minimality of two-bridge knots (2017)
  • Author(s): Fumikazu Nagasato, Masaaki Suzuki, and Anh Tran
  • Journal Title: International Journal of Mathematics Volume: 28
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Epimorphisms between $2$-bridge knot groups and their crossing numbers (2017)
  • Author(s): Masaaki Suzuki
  • Journal Title: Algebraic and Geometric Topology Volume: 印刷中
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Monodromy maps of fibered $2$-bridge knots as elements in Automorphism of free groups (2016)
  • Author(s): Hiroshi Goda and Masaaki Suzuki
  • Journal Title: Kyungpook Mathematical Journal Volume: 56 Pages: 927-938
  • Peer Reviewed
• [Presentation] On a theorem of Friedl and Vidussi (2022)
  • Author(s): 鈴木正明
  • Organizer: トポロジーとコンピュータ
• [Presentation] Genera and crossing numbers of 2-bridge knots (2022)
  • Author(s): 鈴木正明
  • Organizer: 日本数学会
• [Presentation] Genera and crossing numbers of 2-bridge knots (2022)
  • Author(s): 鈴木正明
  • Organizer: 拡大KOOKセミナー
• [Presentation] Knot group, symmetric group, dihedral group, and twisted Alexander polynomial (2022)
  • Author(s): Masaaki Suzuki
  • Organizer: East Asian Conference on Geometric Topology
  • Int'l Joint Research / Invited
• [Presentation] Epimorphisms between knot groups and twisted Alexander polynomial (2022)
  • Author(s): Masaaki Suzuki
  • Organizer: Topology and Geometry of Low-dimensional Manifolds
  • Int'l Joint Research / Invited
• [Presentation] Generating function on epimorphisms between 2-bridge knot groups (2021)
  • Author(s): Masaaki Suzuki
  • Organizer: East Asian Conference on Geometric Topology 2021
  • Int'l Joint Research / Invited
• [Presentation] Epimorphisms between two-bridge knot groups - crossing number, genus (2020)
  • Author(s): Masaaki Suzuki
  • Organizer: トポロジーとコンピュータ2020
• [Presentation] On a structure of the symplectic derivation Lie algebra of the free Lie algebra (2020)
  • Author(s): Masaaki Suzuki
  • Organizer: Teichmuller Theory: Classical, Higher, Super and Quantum
  • Int'l Joint Research / Invited
• [Presentation] Epimorphisms between knot groups (2020)
  • Author(s): Masaaki Suzuki
  • Organizer: SFB Seminar
  • Int'l Joint Research / Invited
• [Presentation] Two filtrations of the Torelli group (2019)
  • Author(s): Masaaki Suzuki
  • Organizer: East Asian Conference on Geometric Topology 2019
  • Int'l Joint Research / Invited
• [Presentation] Computations of Johnson homomorphisms (2018)
  • Author(s): Masaaki Suzuki
  • Organizer: New Trends in Teichmuler Theory and Mapping Class Groups
  • Int'l Joint Research / Invited
• [Presentation] Genera of two-bridge knots and epimorphisms of their knot groups (2018)
  • Author(s): Masaaki Suzuki
  • Organizer: 日本数学会
• [Presentation] Genera of two-bridge knots and epimorphisms of their knot groups (2018)
  • Author(s): Masaaki Suzuki
  • Organizer: 東北結び目セミナー
• [Presentation] Two filtrations of the Torelli group (2018)
  • Author(s): Masaaki Suzuki
  • Organizer: Topology and Computer 2018
• [Presentation] Epimorphisms between knot groups (2017)
  • Author(s): Masaaki Suzuki
  • Organizer: Seminar Geometry, Topology, Dynamical Systems
  • Place of Presentation: University of Texas at Dallas
  • Year and Date: 2017-03-21
  • Invited
• [Presentation] Epimorphisms between knot groups - crossing number, genus (2017)
  • Author(s): Masaaki Suzuki
  • Organizer: East Asian School of Knots and Related Topics
  • Int'l Joint Research / Invited
• [Presentation] Epimorphisms between two-bridge knot groups and their crossing numbers (2016)
  • Author(s): Masaaki Suzuki
  • Organizer: Fundamental Groups, Representations and Geometric Structures in 3-Manifold Topology
  • Place of Presentation: Hiroshima University
  • Year and Date: 2016-11-22
  • Int'l Joint Research / Invited
• [Presentation] Epimorphisms between knot groups (2016)
  • Author(s): Masaaki Suzuki
  • Organizer: Geometry Seminar
  • Place of Presentation: University of Pau
  • Year and Date: 2016-11-15
  • Invited
• [Presentation] Epimorphisms between knot groups (2016)
  • Author(s): Masaaki Suzuki
  • Organizer: International Conference for the 70th Anniversary of Korean Mathematical Society
  • Place of Presentation: Seoul National University
  • Year and Date: 2016-10-22
  • Int'l Joint Research / Invited