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Abstract We introduce a multiparameter deformation of the triply‐graded Khovanov–Rozansky homology of links colored by one‐column Young diagrams, generalizing the “y‐ified” link homology of Gorsky–Hogancamp and work of Cautis–Lauda–Sussan. For each link component, the natural set of deformation parameters corresponds to interpolation coordinates on the Hilbert scheme of the plane. We extend our deformed link homology theory to braids by introducing a monoidal dg 2‐category of curved complexes of type A singular Soergel bimodules. Using this framework, we promote to the curved setting the categorical colored skein relation from our recent joint work and also the notion of splitting map for the colored full twists on two strands. As applications, we compute the invariants of colored Hopf links in terms of ideals generated by Haiman determinants and use these results to establish general link splitting properties for our deformed, colored, triply‐graded link homology. Informed by this, we formulate several conjectures that have implications for the relation between (colored) Khovanov–Rozansky homology and Hilbert schemes.
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Algebra and geometry of link homology: Lecture notes from the IHES 2021 Summer School
Abstract These notes cover the lectures of the first named author at 2021 IHES Summer School on “Enumerative Geometry, Physics and Representation Theory” with additional details and references. They cover the definition of Khovanov‐Rozansky triply graded homology, its basic properties and recent advances, as well as three algebro‐geometric models for link homology: braid varieties, Hilbert schemes of singular curves and affine Springer fibers, and Hilbert schemes of points on the plane.
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- Award ID(s):
- 1760329
- PAR ID:
- 10420130
- Publisher / Repository:
- Oxford University Press (OUP)
- Date Published:
- Journal Name:
- Bulletin of the London Mathematical Society
- Volume:
- 55
- Issue:
- 2
- ISSN:
- 0024-6093
- Page Range / eLocation ID:
- p. 537-591
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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