2005 Fiscal Year Final Research Report Summary
STUDY OF TOPOLOGICAL INVARIANTS FOR KNOTS AND THEIR APPLICATIONS
| Project/Area Number |
14340027
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Osaka City University |
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Principal Investigator |
KANENOBU Taizo Osaka City University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (00152819)
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| Co-Investigator(Kenkyū-buntansha) |
KAWAUCHI Akio Osaka City University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (00112524)
KOMORI Youhei Osaka City Univ., Graduate School of Science, Ass.Professor, 大学院・理学研究科, 助教授 (70264794)
KAMADA Seiichi Hiroshima University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (60254380)
NAKANISHI Yasutaka Kobe University, Faculty of Science, Professor, 理学部, 教授 (70183514)
OHYAMA Yoshiyuki Tokyo Women's Christian Univ., College of Arts and Science, Prof., 理学部, 教授 (80223981)
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| Project Period (FY) |
2002 – 2005
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| Keywords | knot / link / Cn-move / finite type invariant / HOMFLY polynomial / Links-Gould polynomial / Kauffman polynomial / virtual knot |
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| Research Abstract |
There are five main results.
1.We investigated how a delta move influences the first HOMFLY coefficient polynomials of a link. Then we generalized this to a Cn-move.
2.We studied the Links-Gould (LG) polynomial, which is a quantum invariant. Using a skein relation discovered by Ishii, we found a way to construct knots or links sharing the same LG polynomial. Then we gave arbitrarily many 2-bridge knots and links with the same LG polynomial. These 2-bridge knots and links also share the same HOMFLY, Kauffman, and 2-variable Alexander (in case of links) polynomials.
3.We give formulas for the first four coefficient polynomials of the Kauffman's link polynomial involving linking numbers and the coefficient polynomials of the Kauffman polynomials of the one- and two-component sublinks.
4.Giving a presentation of the group of a 2-braid virtual knot or link, we consider the groups of certain special families of 2-braid virtual knots. It is known that the collection of the virtual knot groups is the same as that of the ribbon torus-knot groups. Using our examples we discuss the relationship among the virtual knot groups and other knot groups such as ribbon 2-sphere-knot groups, 2-sphere-knot groups, torus-knot groups, and 3-sphere-knot groups.
5.We give a skein relation for the HOMFLYPT polynomials of 2 cable links. Using this, we have shown that the collection of 2-bridge knots or links mentioned in part 2 also share the same 2-cable HOMFLY polynomial.
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Research Products
(21 results)
