| Project/Area Number |
17500014
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Fundamental theory of informatics
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| Research Institution | Nihon University |
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Principal Investigator |
TANI Seiichi Nihon University, College of Humanities and Sciences, Professor (70266708)
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| Co-Investigator(Kenkyū-buntansha) |
YAKU Takeo Nihon University, College of Humanities and Sciences, Professor (90102821)
TODA Seinosuke Nihon University, College of Humanities and Sciences, Professor (90172163)
HARA Masao Tokai University, Department of Mathematical Science, Associate Professor (10238165)
山本 慎 中央大学, 理工学部, 教授 (10158305)
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| Project Period (FY) |
2005 – 2007
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| Project Status |
Completed (Fiscal Year 2007)
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| Budget Amount *help |
¥3,440,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥240,000)
Fiscal Year 2007: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2006: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2005: ¥1,400,000 (Direct Cost: ¥1,400,000)
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| Keywords | Computational topoloay / Jones polynomial / discrete al aorithm / computational complexity / knot theory / 多項式不変量 / アルゴリズム / トポロジー / 量子計算 |
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| Research Abstract |
Knot theory is a subfield of topology. A knot is a simple (non-self-intersecting) closed curve embedded in R^3. More generally, one may study links. A link is a finite collection of disjointly embedded knots. Works on knot theory have led to many important advances in other areas, biology, chemistry and physics. For classifying and characterizing links, various invariants have been defined and profoundly studied in knot theory. The Jones polynomial is a useful invariant. It is known that computing the Jones polynomial is generally #P-hard. It is expected to require exponential time in the worst case. Recently, it has been recognized that it is important to compute Jones polynomials for links with reasonable restrictions.
EIn this project, we showed that Jones polynomials of pretzel links are computable in O(n^2) time, where n is the number of the edges in the input Tait graph. Moreover, we propose a fast algorithm for computing Jones polynomials of Montesinos links. Given the Tait graph of a Montesinos diagram with n edges, our algorithm runs with O(n) additions and multiplications in polynomials of degree O(n), namely in O(n^2log {n}) time.
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