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Energy of knots (existence of energy minimizers and numerical experiment)

Research Project

Project/Area Number 10640085
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionTokyo Metropolitan University

Principal Investigator

KURATA Toshihiko (2000-2001)  Tokyo Metoropolitan University, Graduate school of Science, 理学研究科, 助手 (40311899)

今井 淳 (1998-1999)  東京都立大学, 大学院・理学研究科, 助教授 (70221132)

Co-Investigator(Kenkyū-buntansha) OKA Mutsuo  Tokyo Metropolitan University, Graduate school of Science, 理学研究科, 教授 (40011697)
OHNITA Yoshihiro  Tokyo Metropolitan University, Graduate school of Science, 理学研究科, 教授 (90183764)
IMAI Jun  Tokyo Metropolitan University, Graduate school of Science, 理学研究科, 助教授 (70221132)
DEGUCHI Tetsuo  Ochanomizu university, Faculty of Science, 理学部, 助教授 (70227544)
NAKAMULA Ken  Tokyo Metropolitan University, Graduate school of Science, 理学研究科, 教授 (80110849)
阿原 一志  明治大学, 理工学部, 専任講師 (80247147)
荻上 紘一  東京都立大学, 総長 (10087025)
Project Period (FY) 1998 – 2001
Project Status Completed (Fiscal Year 2001)
Budget Amount *help
¥2,800,000 (Direct Cost: ¥2,800,000)
Fiscal Year 2001: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 2000: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 1999: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 1998: ¥1,200,000 (Direct Cost: ¥1,200,000)
Keywordstopology / knot / energy / 低次元トポロジー / 結び目のエネルギー
Research Abstract

Roughly speaking, energy of knots is a functional on the space of knots which fulfills the property that it explodes whenever a knot degenerates to a singular knot with double points. During the period of the project, we have studied this concept of energy functional together with Prof. R. Langevin of Universit de Bourgogne mainly to clarify the correspondence between our introduced functional E and the one due to Prof. Langevin which is contrived from the viewpoint of integral geometry. As a result, considering the notion of infinitesimal cross-ratio, we described the energy E in terms of its real-part and its absolute value, and also obtained a formula which describes Langevin's introduced quantity. Furthermore, we newly introduced several conformally invariant functional from a discussion independent of the approach of infinitesimal cross-ratio and integral geometry.
Besides the observation above, we also direct our attention to a strong analogy between the assignment of algebraic invariant of knots and the assignment of denotation of programs containing recursive-calls, and especially study the general mechanism of denotational semantics of cyclic structures appearing in the syntax of programs. In consequence, we obtained an algebraic structure which enables us to model not only the computational paradigm of (type-free) λ-calculus but the internal structure of functional body. Our construction can be regarded as enough simple in comparison with some known results, and so we expect to find a neat relationship to the discussion of energy of knots through a general framework, such as the theory of traced monoidal categories.

Report

(5 results)
  • 2001 Annual Research Report   Final Research Report Summary
  • 2000 Annual Research Report
  • 1999 Annual Research Report
  • 1998 Annual Research Report
  • Research Products

    (60 results)

All Other

All Publications (60 results)

  • [Publications] M.Mukai, Hidano, Y.Ohnita: "Geometry of the modul : spaces of harmonic maps into Lie groups via gange theory over Riemann surfaces"International J.Math.. 12. 339-371 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M.Mukai, Y.Ohnita: "Gange-theoritic approach to harmonic maps and sub spaces in moduli spaces"to appear in "Integral systems, Geometry and Topology".

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Y.Nishimori, Y.Ohnita: "Quantum cohomology rings of Hermition Symmetric spaces of type DI II"to appear in the proceedings of 2001 "WSES International Conference on Applied and Theoretical Math".

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] A.Amarzaya, Y.Ohnita: "On Hamiltonian stability of certain H.minimal Lagragian submanifolds in Hermition symmetric spaces"数理解析研究所講究録. 1236. 31-48 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Y.Ohnita: "Harmonic maps into symmetric spaces and integrasle system theory"上智大学微分幾何学研究会"Lie Groups and Manifold theory"報告集.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M.Oka: "Elliptic Curves from sextics"To appear in J.Math.Soc.Japan.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M.Oka: "Another Involution on moduli of sextics"Kodai J.Math.. 24. 26-30 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M.Oka, D.T.Pho: "Fundamental group of sextics of torus type"To appear in Trends in Singularities.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Kurata: "Intersection and singleton type assigumeur characterizing finite Bohur-trees"To appear in Information and Computation.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Ishihara, T.Kurata: "Completeness of intersection and union type assignment systems for call-by-value Q-models"Theoretical Computer Science. 272. 197-221 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M.K.Shimamura, T.Deguchi: "Topological entropy of a stiff ring polymer and its connection to DNA knots"J.Phys.Soc.Jpn.. 70. 1523-1536 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Deguchi, P.K.Ghosh: "Spin chains from super-models"J.Phys.Soc.Jpn.. 70. 3225-3237 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M.K.Shimamura, T.Deguchi: "Gyration radius of a circular polymer under a topological constraint with excluded volume"Phys.Rev.. 64. (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Deguchi: "Non-regular eigenstate of the XXX model as some limit of the Bethe state"J.Phys.A : Math.Gen.. 34. 9755-9775 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Deguchi: "Construction of some wissing eigenvectors of the XYZ spin chain at the digcrete coupling constants and the exponentially large spectral degeneracy of the transfer matrix"J.Phys.A : Math.Gen.. 35. 879-895 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.M.Bahig, H.El.Zahar, K.Nakamula: "Some results for some conjectures in addition chains"In proceedings of the third International Conference on Continaterics Computability and Legis. 47-54 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.M.Bahig, K.Nakamula: "Some properties of non star steps in addition chains and new cases where the Scholz conjecture is true"to appear in J.Algorithms.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Deguchi, K.Fabricius, B.M.Mccog: "The Sl_2 loop algebra symmetry of the six vertex model at roots of unity"J.Stat.Phys.. 102. 701-701 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] A.Yao, H.Matsuda, H.Tsukahara, M.K.Shimamura, T.Deguchi: "On the dominance of rvial knouts among SAPs on a Cubic lattice"J.Phys.A : Math.Geu.. 34. 7563-7577 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M. Mukai-Hidano and Y. Ohnita: "Geometry of the moduli spaces of harmonic maps into Lie groups via gauge theory over Riemann surfaces"International J. Math.. 12. 339-371 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M. Mukai and Y. Ohnita: "Gauge-theoretic approach to harmonic maps and subspaces in moduli spaces, (a survey article)""Integrable systems, Geometry and Topology", NCTS (National Center for Theoretical Sciences, Taiwan) volume edited by C.-L. Terng, International Press.. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Y. Nishimori and Y. Ohnita: "Quantum cohomology rings of Hermitian symmetric spaces of type DIII"the proceedings of 2001 "WSES International Conference on Appied and Theoretical Mathematics", Cairn, Queensland, Australia, Dec.17-21. (to appear). (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M. Oka: "Elliptic Curves from sextics"J. Math. Soc. Japan. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M. Oka: "Another involution on moduli of sextics"Kodai J Math.. Vol. 24, No.1. 26-30 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M. Oka and D.T. Pho: "Fundamental group of sextics of torus type"Trends in Singularities, Editors A. Libgober and M. Tibar, Birkhauser. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T. Kurata: "Intersection and singleton type assignment characterizing finite Bohm-trees"Information and Computaion. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H. Ishihara and T. Kurata: "Completeness of intersection and union type assignment systems for call-by-value λ-models"Theoretical Computer Science. 272. 197-221 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Tetsuo Deguchi, Klaus Fabricius and Barry M. McCoy: "The sl_2 loop algebra symmetry of the six-vertex model at roots of unity"J. Stat. Phys.. 102. 701 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Miyuki K. Shimamura, and Tetsuo Deguchi: "Topological entropy of a stiff ring polymer and its connection to DNA knots"J. Phys. Soc. Jpn.. 70. 1523-1536 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Tetsuo Deguchi, and Pijush K. Ghosh: "Spin chains from super-models"J. Phys. Soc. Jpn.. 70 No.11. 3225-3237 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Miyuki K. Shimamura, and Tetsuo Deguchi: "Gyration radius of a circular polymer under a topological constraint with excluded volume"Phys. Rev. E. 64. 020801(R) (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Akihisa Yao, Hiroshi Matsuda, Hiroshi Tsukahara, Miyuki K. Shimamura, and Tetsuo Degnchi: "On the dominance of trvial knots among SAPs on a cubic lattice"J. Phys. A : Math. Gen.. 34. 7563-7577 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Tetsuo Degachi: "Non-regular eigenstate of the XXX model as some limit of the Bethe state"J. Phys. A : Math. Gen.. 34. 9755-9775 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Tetsuo Degachi: "Construction of some missing eigenvectors of the XYZ spin chain at the discrete coupling constants and the exponentially large spectral degeneracy of the transfer matrix"J. Phys. A : Math. Gen.. 35. 879-895 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Hatem M. Bahig, H. El-Zahar, and Ken Nakamula: "Some results for some conjectures in addition chains"In Combinatrics, Computability and Logic, Proceedings of the Third International Conference on Combinatrics, Computability and Logic, (DMTCS'01), 47-54, Ovidius University Constantza, Romania, July 2001. Springer Verlag.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Hatem M. Bahig and Ken Nakamula: "Some properties of nonstar steps in addition chains and new cases where the Scholz conjecture is true"J. Algorithms. (accepted). (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M.Mukai-Hidano, Y.Ohnita: "Geometry of the moduli spaces of harmonic maps into Lie groups via gauge theory over Riemann surfaces"International J. Math.. 12. 339-371 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] M.Mukai, Y.Ohnita: "Gauge-theoretic approach to harmonic maps and subspaces in moduli spaces"To appear in "Integrasle systems, Geometry and Topology".

    • Related Report
      2001 Annual Research Report
  • [Publications] Y.Nishimori, Y.Ohnita: "Quantum cohomology rings of Hermitian symmetric spacer of type DIII"To appear in the proceedings of 2001 "WSES International conference on applied and theoretical mash.".

    • Related Report
      2001 Annual Research Report
  • [Publications] A.Amarzaya, Y.Ohnita: "On Hamiltonian stability of certain H-minimal Lagrangian Submanifolds in Hermitian symmetric spaces"数理解析研究所講究録. 1236. 31-48 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Y.Ohnita: "Harmonic maps into symmetric spaces and integrasle system theory"上智大学微分幾何学研究会"Lie Groups and Manifold theory"報告集. (掲載予定).

    • Related Report
      2001 Annual Research Report
  • [Publications] M.Oka: "Elliptic curues from sextics"J. Math. Soc. Japan. (to appear).

    • Related Report
      2001 Annual Research Report
  • [Publications] M.Oka: "Another involution on moduli of sextics"Kodai J. Math.. 24. 26-30 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] M.Oka, D.T.Pho: "Fundamental group of sextics of torus type"Trends in Siugularities. (to appear).

    • Related Report
      2001 Annual Research Report
  • [Publications] T.Kurata: "Intersection and singleson type assignment aharacteriting finite Rohm-trees"Information and Computation. (to appear).

    • Related Report
      2001 Annual Research Report
  • [Publications] H.Ishihara, T.kurata: "Completeness of intersection and union type assignment systems for call-by-value λ-models"Theoretical Computer Science. 272. 197-221 (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] M.K.Shimamura, T.Deguchi: "Topological entropy of a stiff ring polymer and its connection to DNA knots"J. Phys. Soc. Jpn.. 70. 1523-1536 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] T.Deguchi, P.K.Ghosh: "Spin chains from super-models"J. Phys. Soc. Jpn.. 70. 3225-3237 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] M.K.Shimamura, T.Deguchi: "Gyration radius of a circular polymer under a topological constraint with excluded volume"Phys. Rev.. 64. (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] T.Deguchi: "Non-regular eigenstate of the XXX model as some limit of the Bethe state"J. Phys. A : Math. Gen.. 34. 9755-9775 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] T.Deguchi: "Construction of some missing eigenvectors of the XYZ spin chain at the discrete coupling constants and the exponentially large spectral degeneracy of the transfer matrix"J. Phys. A : Math. Gen.. 35. 879-895 (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] H.M.Bahig, H.El-Zahar, K.Nakamula: "S-me results for some conjectures in addition chains"In Proceedings of the thirel International Conference on Conbinatrics Computasiliy and Logic. 47-54 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] H.M.Bahig, K.Nakaumla: "Some properties of nonstar steps in addition chains and new cases where the scholz conjecture is true"J. Algorithms. (to appear).

    • Related Report
      2001 Annual Research Report
  • [Publications] T.Deguchi, K.Fabricius, B.M.McCoy: "The sl_2 loop algebra symmetry of the six-vertex model at roots of unity"J. Stat. Phys.. 102. 701-701 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] A.Yao, H.Matsuda, H.Tsukahara, M.K.Shimamura, T.Deguchi: "On the dominance of trivial knots among SAPs on a cubic lattice"J. Phys. A : Math. Gen.. 34. 7563-7577 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Toshihiko Kurata: "Intersection and singleton type assignment characterizing finite Bolus-trees"Information and Computation. (To appear).

    • Related Report
      2000 Annual Research Report
  • [Publications] Hajime Ishihara Toshihiko Kurata: "Completeness of intersection and union type assignment systems for call-by・value λ-models"Theoretical Computer Science. (To appear).

    • Related Report
      2000 Annual Research Report
  • [Publications] Jun O'Hara: "Asymptotic formulae of energies of polygonal knots"Proceedings of the conference on Low Dimensional Topology, H.Nencka & R.Vasconcelos eds. Contemporary Mathemihcs book series. 235-249 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Jun O, Hara: "Energy of knots" "Ideal Knots" Stasiak, Katrich, Kanffman eds World Scientific. 288-314 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] Jun O, Hara: "Asymptotic formulae of energies of polygonal knots" to appear in Proceedings of the Conference on Low Dimensional Topology Nenclsa & Vasconcelos eds. CONM book series.

    • Related Report
      1998 Annual Research Report

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Published: 1998-04-01   Modified: 2016-04-21  

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