Computing cobordism maps in link Floer homology and the reduced Khovanov TQFT
We study the maps induced on link Floer homology by elementary decorated link cobordisms. We compute these for births, deaths, stabilizations, and destabilizations, and show that saddle cobordisms can be computed in terms of maps in a decorated skein exact triangle that extends the oriented skein ex...
| Main Authors: | , |
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| Format: | Journal article |
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Springer
2017
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| _version_ | 1797067876397481984 |
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| author | Juhasz, A Marengon, M |
| author_facet | Juhasz, A Marengon, M |
| author_sort | Juhasz, A |
| collection | OXFORD |
| description | We study the maps induced on link Floer homology by elementary decorated link cobordisms. We compute these for births, deaths, stabilizations, and destabilizations, and show that saddle cobordisms can be computed in terms of maps in a decorated skein exact triangle that extends the oriented skein exact triangle in knot Floer homology. In particular, we completely determine the Alexander and Maslov grading shifts. As a corollary, we compute the maps induced by elementary cobordisms between unlinks. We show that these give rise to a (1+1)-dimensional TQFT that coincides with the reduced Khovanov TQFT. Hence, when applied to the cube of resolutions of a marked link diagram, it gives the complex defining the reduced Khovanov homology of the knot. Finally, we define a spectral sequence from (reduced) Khovanov homology using these cobordism maps, and we prove that it is an invariant of the (marked) link. |
| first_indexed | 2024-03-06T22:02:40Z |
| format | Journal article |
| id | oxford-uuid:4f1f2555-7995-4c57-8e20-b7cb0ac2f200 |
| institution | University of Oxford |
| last_indexed | 2024-03-06T22:02:40Z |
| publishDate | 2017 |
| publisher | Springer |
| record_format | dspace |
| spelling | oxford-uuid:4f1f2555-7995-4c57-8e20-b7cb0ac2f2002022-03-26T16:05:15ZComputing cobordism maps in link Floer homology and the reduced Khovanov TQFTJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:4f1f2555-7995-4c57-8e20-b7cb0ac2f200Symplectic Elements at OxfordSpringer2017Juhasz, AMarengon, MWe study the maps induced on link Floer homology by elementary decorated link cobordisms. We compute these for births, deaths, stabilizations, and destabilizations, and show that saddle cobordisms can be computed in terms of maps in a decorated skein exact triangle that extends the oriented skein exact triangle in knot Floer homology. In particular, we completely determine the Alexander and Maslov grading shifts. As a corollary, we compute the maps induced by elementary cobordisms between unlinks. We show that these give rise to a (1+1)-dimensional TQFT that coincides with the reduced Khovanov TQFT. Hence, when applied to the cube of resolutions of a marked link diagram, it gives the complex defining the reduced Khovanov homology of the knot. Finally, we define a spectral sequence from (reduced) Khovanov homology using these cobordism maps, and we prove that it is an invariant of the (marked) link. |
| spellingShingle | Juhasz, A Marengon, M Computing cobordism maps in link Floer homology and the reduced Khovanov TQFT |
| title | Computing cobordism maps in link Floer homology and the reduced Khovanov TQFT |
| title_full | Computing cobordism maps in link Floer homology and the reduced Khovanov TQFT |
| title_fullStr | Computing cobordism maps in link Floer homology and the reduced Khovanov TQFT |
| title_full_unstemmed | Computing cobordism maps in link Floer homology and the reduced Khovanov TQFT |
| title_short | Computing cobordism maps in link Floer homology and the reduced Khovanov TQFT |
| title_sort | computing cobordism maps in link floer homology and the reduced khovanov tqft |
| work_keys_str_mv | AT juhasza computingcobordismmapsinlinkfloerhomologyandthereducedkhovanovtqft AT marengonm computingcobordismmapsinlinkfloerhomologyandthereducedkhovanovtqft |