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| Main Authors: | , , , |
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| Format: | Preprint |
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2022
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| Online Access: | https://arxiv.org/abs/2210.00878 |
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| _version_ | 1866917894069157888 |
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| author | Beliakova, Anna Putyra, Krzysztof K. Robert, Louis-Hadrien Wagner, Emmanuel |
| author_facet | Beliakova, Anna Putyra, Krzysztof K. Robert, Louis-Hadrien Wagner, Emmanuel |
| contents | In 2005 Dunfield, Gukov and Rasmussen conjectured an existence of the spectral sequence from the reduced triply graded Khovanov-Rozansky homology of a knot to its knot Floer homology defined by Ozsváth and Szabó. The main result of this paper is a proof of this conjecture. For this purpose, we construct a bigraded spectral sequence from the $\mathfrak{gl}_0$ homology constructed by the last two authors to the knot Floer homology. Using the fact that the $\mathfrak{gl}_0$ homology comes equipped with a spectral sequence from the reduced triply graded homology, we obtain our main result. The first spectral sequence is of Bockstein type and comes from a subtle manipulation of coefficients. The main tools are quantum traces of foams and of singular Soergel bimodules and a $\mathbb Z$-valued cube of resolutions model for knot Floer homology originally constructed by Ozsváth and Szabó over the field of two elements. As an application, we deduce that the $\mathfrak{gl}_0$ homology as well as the reduced triply graded Khovanov-Rozansky one detect the unknot, the two trefoils, the figure eight knot and the cinquefoil. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_00878 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A proof of Dunfield-Gukov-Rasmussen Conjecture Beliakova, Anna Putyra, Krzysztof K. Robert, Louis-Hadrien Wagner, Emmanuel Geometric Topology Algebraic Topology 57K18, 18G40, 55U20 In 2005 Dunfield, Gukov and Rasmussen conjectured an existence of the spectral sequence from the reduced triply graded Khovanov-Rozansky homology of a knot to its knot Floer homology defined by Ozsváth and Szabó. The main result of this paper is a proof of this conjecture. For this purpose, we construct a bigraded spectral sequence from the $\mathfrak{gl}_0$ homology constructed by the last two authors to the knot Floer homology. Using the fact that the $\mathfrak{gl}_0$ homology comes equipped with a spectral sequence from the reduced triply graded homology, we obtain our main result. The first spectral sequence is of Bockstein type and comes from a subtle manipulation of coefficients. The main tools are quantum traces of foams and of singular Soergel bimodules and a $\mathbb Z$-valued cube of resolutions model for knot Floer homology originally constructed by Ozsváth and Szabó over the field of two elements. As an application, we deduce that the $\mathfrak{gl}_0$ homology as well as the reduced triply graded Khovanov-Rozansky one detect the unknot, the two trefoils, the figure eight knot and the cinquefoil. |
| title | A proof of Dunfield-Gukov-Rasmussen Conjecture |
| topic | Geometric Topology Algebraic Topology 57K18, 18G40, 55U20 |
| url | https://arxiv.org/abs/2210.00878 |