Fray functors and equivalence of colored HOMFLYPT homologies
Abstract
We construct several families of functors on the homotopy category of singular Soergel bimodules that mimic cabling and insertion of column-colored projectors. We use these functors to identify the intrinsically-colored homology of Webster--Williamson and the projector-colored homology of Elias--Hogancamp for an arbitrary link, up to multiplication by a polynomial in the quantum degree $q$. Combined with the results of arXiv:2303.16271, this establishes parity results for the intrinsic column-colored homology of positive torus knots, partially resolving a conjecture of Hogancamp--Rose--Wedrich in arXiv:2107.09590.
- Publication:
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arXiv e-prints
- Pub Date:
- May 2024
- DOI:
- 10.48550/arXiv.2405.00875
- arXiv:
- arXiv:2405.00875
- Bibcode:
- 2024arXiv240500875C
- Keywords:
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- Mathematics - Quantum Algebra;
- Mathematics - Geometric Topology
- E-Print:
- 79 pages, many figures. Comments and suggestions welcome!