2004 Fiscal Year Final Research Report Summary
On computational algorithms of invariatns of links and graphs
| Project/Area Number |
14540136
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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 |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | CHUO UNIVERSITY |
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Principal Investigator |
YAMAMOTO Makoto Chuo University, Faculty of Science and Engineering, Professor, 理工学部, 教授 (10158305)
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| Co-Investigator(Kenkyū-buntansha) |
MATSUYAMA Yoshio Chuo University, Faculty of Science and Engineering, Professor, 理工学部, 教授 (70112753)
MIYOSHI Shigeaki Chuo University, Faculty of Science and Engineering, Professor, 理工学部, 教授 (60166212)
MITSUMATSU Yoshihiko Chuo University, Faculty of Science and Engineering, Professor, 理工学部, 教授 (70190725)
SEKIGUCHI Tsutomu Chuo University, Faculty of Science and Engineering, Professor, 理工学部, 教授 (70055234)
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| Project Period (FY) |
2002 – 2004
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| Keywords | knot / link / topological invariant / Jones polynomial / interactive proof system / computational topology |
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| Research Abstract |
The purpose of this research is to study alogrithms of cmuting topologizal invariants of knots, links and graphs.
Our results are followings:
1. (2003) Let L be a link, c the number of crossings of a diagram of L. We showed that the Jones polynomial of an arborescent link is computed with O(c^3) operations of polynomials of degree O(c).
2. (2004) We constructed an interactive proof system for the Knotting Problem, and proved that the problem is contained in IP. Consequently, the Unknotting Problem is contained in both AM and co-AM.
3. (2004) We gave 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), where n is the number of the crossings of the link diagram.
4. (2005) We gave 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 in O(n^2log n) time.
5. (2005) We gave 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).
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