2007 Fiscal Year Final Research Report Summary
Smooth unknotting conjecture in dimension four and search for essence of mathematics
| Project/Area Number |
16340017
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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 | Hiroshima University |
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Principal Investigator |
MATUMOTO Takao Hiroshima University, Graduate School of Science, Professor (50025467)
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| Co-Investigator(Kenkyū-buntansha) |
KAMADA Seiichi Hiroshima University, Graduate School of Science, Professor (60254380)
MATSUMOTO Yukio Gakushuin University, Faculty of Science, Professor (20011637)
UENO Kenji Kyoto University, Graduate School of Science, Professor (40011655)
WAKAKI Hirofumi Hiroshima University, Graduate School of Science, Professor (90210856)
NAGAI Toshitaka Hiroshima University, Graduate School of Science, Professor (40112172)
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| Project Period (FY) |
2004 – 2007
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| Keywords | Topologv / Knot / Braid / Chart description / Cusp / 1-parameter family / Elementary method / Historical search |
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| Research Abstract |
The necessary and. ant condition for unknotting is well-known except for a smooth 2-knot. The complement has the same homotopy type with a trivial knot is the condition. Our aim is to show that this condition is enough also for the remaining case, using 2-dimensional braid theory The method is quite elementary and we want to find some essence of mathematics in this chance.
1) We have already a one-parameter family of maps with only cusp births and deaths between a given knot with the condition and a trivial knot. 2) Transform this to a one-parameter family of singular 2-braids by extending a method due to Kamada. 3) Find its chart description by applying stabilization if necessary. 4) The height of an intersection point can be modified down until the level a little higher than the related cusp death. 5) This reduces the problem to the case that the number of intersection points is one and the intersection point go down by rotating the other fixed vertices of the chart and then disappear at the cusp. 6) This step can be divided into several simple steps by a word representation due to Kamada-Matsumoto. 7) Each step will be shown that the ends are turned out to be trivial 2-dimensional braids. This is an outline to solve the conjecture.
As for essence of mathematics besides the study by each investigator we studied the history of determinants at the occasion of 300 years after the death of Takakazu Seki.
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