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2000 Fiscal Year Final Research Report Summary

Invariants for links in thickened surfaces and its applications

Research Project

Project/Area Number 11640059
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionUniversity of Tsukuba

Principal Investigator

KANETO Takeshi  University of Tsukuba, Institute of Mathematics, lecturer, 数学系, 講師 (70107340)

Co-Investigator(Kenkyū-buntansha) KAWAMURA Kazuhiro  University of Tsukuba, Institute of Mathematics, assistant professor, 数学系, 助教授 (40204771)
SAKAI Katsuro  University of Tsukuba, Institute of Mathematics, assistant professor, 数学系, 助教授 (50036084)
KATO Hisao  University of Tsukuba, Institute of Mathematics, professor, 数学系, 教授 (70152733)
YAMAGATA Kunio  Tokyo University of Agriculture and Technology, Department of Mathematics, professor, 工学部, 教授 (60015849)
AKUTAGAWA Reiko (AIYAMA,REIKO)  University of Tsukuba, Institute of Mathematics, lecturer, 数学系, 講師 (20222466)
Project Period (FY) 1999 – 2000
Keywordsthickened surface / knot / link / alternating link / Tait conjecture / bracket / link invariant / minimal crossing number
Research Abstract

Tait conjectures on crossing numbers of alternating links in the 3-dimensional space means the following two conjectures :
TC1) If two connected, reduced and alternating link diagrams represent equivalent (ie, ambient isotopic) links, their crossing numbers will be the same.
TC2) The crossing number of a connected, reduced and alternating link diagram D will be minimal among all crossing numbers of link diagrams which represent links equivalent to that D does.
These conjectures had been unsolved for more than 100 years untill they were positively solved in 1987 after Jones polynomial appeared. In 1996, N.Kamada made a pioneer-work to establish anologous results for alternating links in thickened surfaces. She showed an anologous result to TC1) under a certain assumption and conjectured that the assumption can be removed in her paper. In this research project, we solved her conjecture positively . And, by applying the idea of the proof of her conjecture, we established an anologous result … More to TC2), which is a final goal to seeking anologous results to Tait conjectures in knot-link theory in thickened surfaces because TC2) implies TC1. Our these results was showed for more general both surfaces and link diagrames than those in her paper. In the proofs of our these results, we use the following two fundamental lemmta :
ENGULFING LEMMA : For two link diagrams on a surface, one of their regular neiborhoods in the the surface can contain the other by isotopic deformation in the surface under a certain condition,
DUAL STATE LEMMA (thickened surface version) : For a pair of dual states of a link diagram D on a surface, the sum of numbers of connected components of two (no crossing) diagrams obtained from D by resolving all crossings according to each state does not exceed to the number of boundary components of the regular neighhood of D in the surface under a certain condition.
These lemmata, themselves, are impotant results in our resaerch. Each invesigator got intereting own results as the fundamental research related to this project. Less

  • Research Products

    (36 results)

All Other

All Publications (36 results)

  • [Publications] Takeshi Kaneto: "Tait type theorems on alternating links in thikened surfaces""Knot theory" - dedicated to Professor Murasugi's 70th birthday -(edited by Makoto Sakuma). 148-156 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Hisao Kato: "Expansive homeomorphisms of countable compacta"Topology and its Appl.. 95. 207-216 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Hisao Kato: "Open maps on manifolds which do not admit disjoint closed subsets intersecting each fiber"Topology and its Appl.. 103. 221-228 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Hisao Kato K.Kawamura et al.: "Measures and topological dynamics on Menger manifolds"Topology and its Appl.. 103. 249-282 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Takao Hoshina with K.Yamazaki: "Weak C-embedding, weak P-embedding and product spaces"Topology and its Appl.. (To appear).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Katsuro Sakai: "Spaces of upper semi-continuous multi-valued functions on complete metric spaces"Fund.Math.. 160. 199-218 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Katsuro Sakai: "The completion of metric ANR's and homotopy dense subsets"J.Math.Soc.Japan. 52. 835-846 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Kazuhiro Kawamura: "Total excess and Tits metric for piecewise Riemannian 2-manifolds"Topology and its Appl.. 94. 173-193 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Kazuhiro Kawamura: "A survey on Menger manifold theory-Update"Topology and its Appl.. 101. 83-91 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Banakh with K.Kawamura and K.Sakai: "Direct limits of the Banach-Mazur compacta"Bull.London Math.Soc.. 32. 709-717 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Kaori Yamazaki: "A cardinal generalization of C^*-embedding and its applications"Topology and its Appl.. 108. 137-156 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Kaori Yamazaki: "Extensions of partitions of unity"Topology Proc.23 Summer. (To appear).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Reiko Aiyama: "A global correspondence between CMC-surfaces in S^3 and pairs of non-conformal harmonic maps into S^2"Proc.A.M.S.. 128. 939-941 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Reiko Aiyama: "Kenmotsu type representation formula for surfaces with prescribed mean curvature in the 3-sphere"Tohoku Math.J.. 52. 95-105 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Reiko Aiyama: "Kenmotsu type representation formula for surfaces with prescribed mean curvature in the hyperbolic 3-space"J.Math.Soc.Japan. 52. 877-898 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Hisaaki Fujita: "Global and Krull dimension of quantum Weyl algebras"Journal of Algebra. 216. 405-416 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Y.Ohnuki with K.Yamagata: "Symmetric Hochschild Extension Algebras"Colloq.Math.. 80. 155-174 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] O.Kerner with K.Yamagata: "Auslander-Reiten components containing cones"Representation Theory of Algebras. (To appear).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Takeshi Kaneto: "Tait type theorems on alternating links in thickened surfaces""Knot theory"-dedicated to Professor Murasugi's 70th birthday-(edited by Makoto Ssakuma). 148-156 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Hisao Kato: "Expansive homeomorphisms of countable compacta"Topology and its Appl.. 95. 207-216 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Hisao Kato: "Open maps on manifolds which do not admit disjoint closed subsets intersecting each fiber"Topology and its Appl.. 103. 221-228 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Hisao kato with K.Kawamura et al.: "Measures and topological dynamics on Menger manifolds"Topology and its Appl.. 103. 249-282 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Takao Hoshian and K.Yamazaki: "Week C-embedding, week P-embedding and product spaces"Topology and its Appl. to appear.

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Katsuro Sakai: "Spaces of upper semi-continuous multi-valued functions on complete metric spaces"Fund.Math.. 160. 199-218 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Katsuro Sakai: "The completion of metric ANR's and homotopy dense subsets"J.Math.Soc.Japan. 52. 835-846 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Kazuhiro Kawamura et al.: "Total excess and Tits metric for piecewise Riemannian 2-manifolds"Topology and its Appl.. 94. 173-193 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Kazuhiro Kawamura: "A survey on Menger manifold theory-Update"Topology and its Appl.. 101. 83-91 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T.Banakh, K.Kawamura and K.Sakai: "Direct limits of the Banach-Mazur compacta"Bull.London Math.Sco.. 32. 709-717 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Kaori Yamazaki: "A cardinal generalization of C^*-embedding and its applications"Topology and its Appl.. 108. 137-156 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Kaori Yamazaki: "Extensions of partitions of unity"Topology Proc.23 Summer. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Reiko Aiyama: "A global correspondence between _<CM>C-surfaces in S^3 and pairs of non-conformal harmonic maps into S^2"Proc.A.M.S.. 128. 939-941 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Reiko Aiyama: "Kenmotsu type representation formula for surfaces with prescribed mean curvature in the 3-sphere"Tohoku Math.J.. 52. 95-105 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Reiko Aiyama: "Kenmotsu type representation formula for surfaces with prescribed mean curvature in the hyperbolic 3-space"J.Math.Soc.Japan. 52. 877-898 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Hisaaki Fujita: "Global and Krull dimension of quantum Weyl algebras"J.of Algebra. 216. 405-416 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Y.Ohnuki and K.Yamagata: "Symetric Hochschild Extension Algebra"Colloq.Math.. 80. 155-174 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] O.Kerner and K.Yamagata: "Auslander-Reiten components containg cores"Representation Theory of Algebras. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2002-03-26  

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