2004 Fiscal Year Final Research Report Summary
Developing monodromy descriptions and studying two-dimensional knots and braids by using quandles
| Project/Area Number |
15540077
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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 | HIROSHIMA UNIVERSITY |
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Principal Investigator |
KAMADA Seiichi Hiroshima University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (60254380)
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| Co-Investigator(Kenkyū-buntansha) |
MATUMOTO Takao Hiroshima University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (50025467)
MATSUMOTO Makoto Hiroshima University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (70231602)
TERAGAITO Masakazu Hiroshima University, Graduate School of Education, Associate Professor, 大学院・教育学研究科, 助教授 (80236984)
KAWAUCHI Akio Osaka City University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (00112524)
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| Project Period (FY) |
2003 – 2004
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| Keywords | quandle / biquandle / monodromy / 2-dimensional knot / 2-dimensional braid / chart / knot |
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| Research Abstract |
Investigation on biquandles, which are extension of quandles, is useful for study on knots, two-dimensional knots, and two-dimensional braids. Biquandles of order 4 are classified ; there are 98 types up to isomorphism. The classification can be used in order to determine whether two given biquandles are isomorphic or not. Silver-Williams' invariants of virtual knots are investigated in terms of biquandles.
It is shown that there is a one-to-one correspondence between monodromy representations of any topological objects and elements of enveloping monoidal quandles. Such monodromy representations can be described by charts, and there is a method to find basic moves connecting equivalent charts. Especially, chart description method to describe genus-1 Lefschetz fibrations is established. Using this method, we can classify genus-1 Lefschetz fibrations as fibrations, and classify their total spaces as manifolds. This result was known much earlier when it is chiral and the base space is spherical. Our result gives an elementary proof of a classification theorem of genus-1 Lefschetz fibrations due to Y.Matsumoto. Chart description method to describe genus-2 Lefschetz fibrations is also studied, and it is shown that any genus-2 Lefschetz fibrations can be stabilized, even if they are chiral or achiral. Although, we can obtain chart description method to describe Lefschetz fibrations of any higher genera, we need further study for details. Enveloping monoidal quandles determine associated groups of quandles, and elements of enveloping monoidal quandle correspond to monodromy representations, and we can obtain charts from them.
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Research Products
(9 results)
