Concordance maps in knot Floer homology
We show that a decorated knot concordance C from K to K' induces a homomorphism F_C on knot Floer homology that preserves the Alexander and Maslov gradings. Furthermore, it induces a morphism of the spectral sequences to HF^(S^3) = Z_2 that agrees with F_C on the E^1 page and is the identity on...
| Main Authors: | , |
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| Format: | Journal article |
| Language: | English |
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Mathematical Sciences Publishers
2016
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| _version_ | 1797068094950080512 |
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| author | Juhász, A Marengon, M |
| author_facet | Juhász, A Marengon, M |
| author_sort | Juhász, A |
| collection | OXFORD |
| description | We show that a decorated knot concordance C from K to K' induces a homomorphism F_C on knot Floer homology that preserves the Alexander and Maslov gradings. Furthermore, it induces a morphism of the spectral sequences to HF^(S^3) = Z_2 that agrees with F_C on the E^1 page and is the identity on the E^infinity page. It follows that F_C is non-vanishing on HFK^_0(K, \tau(K)). We also obtain an invariant of slice disks in homology 4-balls bounding S^3.If C is invertible, then F_C is injective, hence dim HFKh_j(K,i) <= dim HFKh_j(K',i) for every i, j in Z. This implies an unpublished result of Ruberman that if there is an invertible concordance from the knot K to K', then g(K) <= g(K'), where g denotes the Seifert genus. Furthermore, if g(K) = g(K') and K' is fibred, then so is K. |
| first_indexed | 2024-03-06T22:05:44Z |
| format | Journal article |
| id | oxford-uuid:5012dff8-f26e-467e-9e48-3d68591f37b5 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-06T22:05:44Z |
| publishDate | 2016 |
| publisher | Mathematical Sciences Publishers |
| record_format | dspace |
| spelling | oxford-uuid:5012dff8-f26e-467e-9e48-3d68591f37b52022-03-26T16:11:27ZConcordance maps in knot Floer homologyJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:5012dff8-f26e-467e-9e48-3d68591f37b5EnglishSymplectic Elements at OxfordMathematical Sciences Publishers2016Juhász, AMarengon, MWe show that a decorated knot concordance C from K to K' induces a homomorphism F_C on knot Floer homology that preserves the Alexander and Maslov gradings. Furthermore, it induces a morphism of the spectral sequences to HF^(S^3) = Z_2 that agrees with F_C on the E^1 page and is the identity on the E^infinity page. It follows that F_C is non-vanishing on HFK^_0(K, \tau(K)). We also obtain an invariant of slice disks in homology 4-balls bounding S^3.If C is invertible, then F_C is injective, hence dim HFKh_j(K,i) <= dim HFKh_j(K',i) for every i, j in Z. This implies an unpublished result of Ruberman that if there is an invertible concordance from the knot K to K', then g(K) <= g(K'), where g denotes the Seifert genus. Furthermore, if g(K) = g(K') and K' is fibred, then so is K. |
| spellingShingle | Juhász, A Marengon, M Concordance maps in knot Floer homology |
| title | Concordance maps in knot Floer homology |
| title_full | Concordance maps in knot Floer homology |
| title_fullStr | Concordance maps in knot Floer homology |
| title_full_unstemmed | Concordance maps in knot Floer homology |
| title_short | Concordance maps in knot Floer homology |
| title_sort | concordance maps in knot floer homology |
| work_keys_str_mv | AT juhasza concordancemapsinknotfloerhomology AT marengonm concordancemapsinknotfloerhomology |