2007 Fiscal Year Final Research Report Summary
Link Theory from the Computational Topological Viewpoint
| Project/Area Number |
17540135
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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 | Tokai University |
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Principal Investigator |
HARA Masao Tokai University, School of Science, Associate Professor (10238165)
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| 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)
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| Project Period (FY) |
2005 – 2007
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| Keywords | link theory / computational topology / Jones polynomial / 2-bridge link / 3-braid link / Montesinos link |
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| 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.
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