2001 Fiscal Year Final Research Report Summary
STUDY ON KNOT INVARIANTS AND ITS APPLICATIONS
| Project/Area Number |
11640090
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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 |
Geometry
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| Research Institution | Osaka City University |
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Principal Investigator |
KANENOBU Taizo Osaka City University, Faculty of Science, Associate Professor, 大学院・理学研究科, 助教授 (00152819)
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| Co-Investigator(Kenkyū-buntansha) |
HASHIMOTO Yoshitake Osaka City University, Faculty of Science, Lecturer, 大学院・理学研究科, 助教授 (20271182)
KAMADA Seiichi Osaka City University, Faculty of Science, Associate Professor, 大学院・理学研究科, 助教授 (60254380)
KAWAUCHI Akio Osaka City University, Faculty of Science, Professor, 大学院・理学研究科, 教授 (00112524)
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| Project Period (FY) |
1999 – 2001
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| Keywords | knot / link / ribbon 2-knot / virtual knot / HC-move / finite type (Vassiliev) invariant / tangle / polynomial invariant |
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| Research Abstract |
We studied on finite type invariants or Vassiliev invariants of ribbon 2-knots, HC-moves for ribbon 2-knots, some properties of HOMFLY polynomials of links, tangle surgeries preserving some polynomial invariants, and the finite type invariants for handcuff graphs.
We defined finite type invariants for a class of ribbon 2-knots. Then we showed that each coefficient in the Taylor expansion of the normalized Alexander polynomial of a ribbon 2-knot is a Vassiliev invariant. There, we constructed a 'Vassiliev-like' filtration in two ways. However, we proved that the two filtrations are the same, and thus, the two finite type invariants are coincident.
We defined the HC-move as an unknotting operation of a ribbon 2-knot as a generalization of a Δ-move for a 1-knot. Then we gave some relatins between the HC-move and the α_2-invariant of a ribbon 2-knot, which is the order 2 finite type invariant. This allowed us to decide the HC-unknotting numbers of some ribbon 2-konts.
Making use of the virtual arc representation of a ribbon 2-knot due to Satoh, we saw that the HC-move corresponds to one of the "forbidden moves", which unknot every virtual knot. Then : (1) We proved that any virtural knot can be unknotted by the forbidden moves. (2) We proved the HC-move is an unknotting operation for the virtual arc representation of a ribbon 2-knot. (3) We gave some relation between the Δ-move for a 1-knot and the HC-move for the spun 2-knot.
We give formulas for the second and third coefficient polynomials of the HOMFLY polynomial of a link which are described by the linking numbers and the coefficient polynomials of the HOMFLY polynomials of the proper sublinks.
We introduce some tangle surgeries on the double of a tangle. If the tangle satisfies certain conditions, then the resulting link has the same polynomial invariant as the original one.
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