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2007 Fiscal Year Final Research Report Summary

Link Theory from the Computational Topological Viewpoint

Research Project

Project/Area Number 17540135
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionTokai University

Principal Investigator

HARA Masao  Tokai University, School of Science, Associate Professor (10238165)

Co-Investigator(Kenkyū-buntansha) ONISHI Kensuke  Tokai University, School of Science, Lecturer (00303024)
MATSUMOTO Satoshi  Tokai University, School of Science, Associate Professor (30307235)
TANI Seiichi  Nihon University, College of Humanities and Sciences, Professor (70266708)
Project Period (FY) 2005 – 2007
Keywordslink theory / computational topology / Jones polynomial / 2-bridge link / 3-braid link / Montesinos link
Research Abstract

Alexander polynomial of a link can be calculated in polynomial time from a link diagram. It is known that calculating Jones polynomial or other skein polynomials from a diagram is #P-hard. Therefore it seems impossible that Jones polynomial is calculated in polynomial time. For some classes of diagrams it is known that we can calculate the Jones polynomial of the links in polynomial time if the class contains its diagram.
In this research, we design a fast algorithm that calculates Kauffman bracket polynomial of a 2-bridge link diagram and a closed 3-braid link diagram. Our algorithm takes O(n) operations of polynomials, where n is the number of crossings of the diagram. Since the degrees of the polynomials that are used are O(n) and the coefficients are O(n) size integers, it takes O(n^2 log n) time word operations. Jones polynomial of a link can be calculated from Kauffinan bracket polynomial and writhe of its diagram in linear time.
Moreover, we show that Kauffinan bracket polynomial of Montesinos link diagrams can be calculated in O(n^2 log n) time. The branched double covering space of the 3-sphere branched along a 2-bridge link or a Montesinos link is a Seifert fiber space whose base space is a 2-sphere. The branched covering space of a 2-bridge link has at most 2 singular fibers and the branched covering space of a Montesinos link has 3 or more singular fibers.

  • Research Products

    (11 results)

All 2007 2005 Other

All Journal Article (6 results) (of which Peer Reviewed: 3 results) Presentation (4 results) Book (1 results)

  • [Journal Article] A Fast Algorithm for Computing Jones Polynomials of Montesinos Links2007

    • Author(s)
      Masao Hara
    • Journal Title

      2007 KOREA-JAPAN Joint Workshop on Algorithms and Computation

      Pages: 40-47

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] Fast algorithms for computing Jones polynomials of certain links2007

    • Author(s)
      Masahiko Murakami
    • Journal Title

      Theoretical Computer Science 374

      Pages: 1-24

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] A Fast Algorithm for Computing Jones Polynomials of Montesinos Links2007

    • Author(s)
      Masao, Hara, Masahiko, Murakami, Seiichi, Tani, Makoto, Yamamoto
    • Journal Title

      2007 KOREA-JAPAN Joint Workshop on Algorithms and Computation

      Pages: 40-47

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Fast algorithms for computing Jones polynomials of certain links2007

    • Author(s)
      Masahiko, Murakami, Masao, Hara, Makoto, Yamamoto, Seiichi, Tani
    • Journal Title

      Theoretical Computer Science 374

      Pages: 1-24

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] The determinants of certain matrices arising from the Boolean lattice

    • Author(s)
      Masao Hara
    • Journal Title

      Discrete Mathematics (印刷中)

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] The determinants of certain matrices arising from the Boolean lattice

    • Author(s)
      Masao, Hara, Junzo, Watanabe
    • Journal Title

      to appear in Discrete Mathematics

    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] A Fast Algorithm for Computing Jones Polynomial of Montesinos Links2007

    • Author(s)
      Masao, Hara, Masahiko, Murakami, Seiichi, Tani, Makoto, Yamamoto
    • Organizer
      Kore-Japan Joint Workshop WAAC 2007
    • Place of Presentation
      Chonnam National University Korea
    • Year and Date
      20070800
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] A Fast Algorithm for Computing Jones Polynomial of Montesinos Links2007

    • Author(s)
      Masao Hara
    • Organizer
      Kore-Japan Joint Workshop WAAC 2007
    • Place of Presentation
      Chonnam National University,Korea
    • Year and Date
      2007-08-09
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] Fast Algorithms for Computing Jones Polynomials of Certain Links2005

    • Author(s)
      Masahiko, Murakami, Masao, Hara, Makoto, Yamamoto, Seiichi, Tani
    • Organizer
      4th Japanese-Hungarian Symposium on Discrete Mathematics and Its Applications
    • Place of Presentation
      Renyi Institute, Budapest Hungary
    • Year and Date
      20050600
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] Fast Algorithms for Computing Jones Polynomials of Certain Links2005

    • Author(s)
      Masahiko Murakami
    • Organizer
      4th Japanese-Hungarian Symposium on Discrete Mathematics and Its Applications
    • Place of Presentation
      Renyi Institute,Budapest,Hungary
    • Year and Date
      2005-06-05
    • Description
      「研究成果報告書概要(和文)」より
  • [Book] Javaではじめるレイトレーシング入門2005

    • Author(s)
      郡山 彬
    • Total Pages
      261
    • Publisher
      森北出版
    • Description
      「研究成果報告書概要(和文)」より

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Published: 2010-02-04  

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