Obstructions to slicing knots and splitting links
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2014.
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| Format: | Thesis |
| Language: | eng |
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Massachusetts Institute of Technology
2014
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| Online Access: | http://hdl.handle.net/1721.1/90178 |
| _version_ | 1811078502318866432 |
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| author | Batson, Joshua |
| author2 | Peter Ozsváth. |
| author_facet | Peter Ozsváth. Batson, Joshua |
| author_sort | Batson, Joshua |
| collection | MIT |
| description | Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2014. |
| first_indexed | 2024-09-23T11:01:11Z |
| format | Thesis |
| id | mit-1721.1/90178 |
| institution | Massachusetts Institute of Technology |
| language | eng |
| last_indexed | 2024-09-23T11:01:11Z |
| publishDate | 2014 |
| publisher | Massachusetts Institute of Technology |
| record_format | dspace |
| spelling | mit-1721.1/901782019-04-11T13:08:42Z Obstructions to slicing knots and splitting links Batson, Joshua Peter Ozsváth. Massachusetts Institute of Technology. Department of Mathematics. Massachusetts Institute of Technology. Department of Mathematics. Mathematics. Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2014. 49 Cataloged from PDF version of thesis. Includes bibliographical references (pages 65-68). In this thesis, we use invariants inspired by quantum field theory to study the smooth topology of links in space and surfaces in space-time. In the first half, we use Khovanov homology to the study the relationship between links in R3 and their components. We construct a new spectral sequence beginning at the Khovanov homology of a link and converging to the Khovanov homology of the split union of its components. The page at which the sequence collapses gives a lower bound on the splitting number of the link, the minimum number of times its components must be passed through one another in order to completely separate them. In addition, we build on work of Kronheimer- Mrowka and Hedden-Ni to show that Khovanov homology detects the unlink. In the second half, we consider knots as potential cross-sections of surfaces in R4. We use Heegaard Floer homology to show that certain knots never occur as cross-sections of surfaces with small first Betti number. (It was previously thought possible that every knot was a cross-section of a connect sum of three Klein bottles.) In particular, we show that any smooth surface in R 4 with cross-section the (2k, 2k - 1) torus knot has first Betti number at least 2k - 2. by Joshua Batson. Ph. D. 2014-09-19T21:44:25Z 2014-09-19T21:44:25Z 2014 2014 Thesis http://hdl.handle.net/1721.1/90178 890207589 eng M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission. http://dspace.mit.edu/handle/1721.1/7582 68 pages application/pdf Massachusetts Institute of Technology |
| spellingShingle | Mathematics. Batson, Joshua Obstructions to slicing knots and splitting links |
| title | Obstructions to slicing knots and splitting links |
| title_full | Obstructions to slicing knots and splitting links |
| title_fullStr | Obstructions to slicing knots and splitting links |
| title_full_unstemmed | Obstructions to slicing knots and splitting links |
| title_short | Obstructions to slicing knots and splitting links |
| title_sort | obstructions to slicing knots and splitting links |
| topic | Mathematics. |
| url | http://hdl.handle.net/1721.1/90178 |
| work_keys_str_mv | AT batsonjoshua obstructionstoslicingknotsandsplittinglinks |