2004 Fiscal Year Final Research Report Summary
On computational complexity of computing polynomial invariants of links
| Project/Area Number |
14580391
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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 |
計算機科学
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| Research Institution | Nihon University |
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Principal Investigator |
TANI Seiichi Nihon University, College of Humanities and Sciences, Associate Professor, 文理学部, 助教授 (70266708)
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| Co-Investigator(Kenkyū-buntansha) |
YAKU Takeo Nihon University, College of Humanities and Sciences, Professor, 文理学部, 教授 (90102821)
TODA Seinosuke Nihon University, College of Humanities and Sciences, Professor, 文理学部, 教授 (90172163)
YAMAMOTO Makoto Chuo University, Faculty of Science and Engineering, Professor, 理工学部, 教授 (10158305)
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| Project Period (FY) |
2002 – 2004
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| Keywords | computational topology / Jones polynomial / discrete algorithms / interactive proof system / knots / links / braid group / conjugacy problem |
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| Research Abstract |
We investigate computational complexity of computing polynomial invariants of links. We also investigate the computational complexity of the problem whether a knot is unknotting and the computational complexity of the computational complexity of the conjugacy problem for braids.
We give fast algorithms for computing Jones polynomials of 2-bridge links and closed 3-braid links from their Tait graphs. Given a Tait graph with n edges, these algorithms run with O(n) arithmetic operations of polynomials of degree O(n) namely in O(n^2log n) time, where n is the number of the crossings of the link diagram. We also give a fast algorithm for computing Jones polynomials of Montesinos links from lists of integer sequences. Given a list of integer sequences that represents a link diagram with $n$ crossings, this algorithm runs with O(n) operations of polynomials of degree O(n).
We construct an interactive proof system for the Knotting Problem, and prove that the problem is contained in IP. Consequently, the Unknotting Problem is contained in both AM and co-AM.
The conjugacy problem for the n-strand braids is the following decision problem : Given two braids V, W, determine whether there exists a braid C such that CV is equivalent to W C. We give a proof that the conjugacy problem for braids is in PSPACE.
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