2021/07/20 by Matthew Hogancamp, Hogancamp, Matthew, David E. V. Rose +3 · 2 citations
Mathematics · #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #FOS: Mathematics #Geometric Topology (math.GT) #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2107.09590
openalex publication_date 2021/07/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce a multi-parameter 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 arXiv:2107.08117 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.