Hilbert schemes and y–ification ofKhovanov–Rozansky homology
Eugene Gorsky, Matthew Hogancamp
Abstract
We define a deformation of the triply graded Khovanov-Rozansky homology of a\nlink $L$ depending on a choice of parameters $y_c$ for each component of $L$,\nwhich satisfies link-splitting properties similar to the Batson-Seed invariant.\nKeeping the $y_c$ as formal variables yields a link homology valued in triply\ngraded modules over $\\mathbb{Q}[x_c,y_c]_{c\\in \\pi_0(L)}$. We conjecture that\nthis invariant restores the missing $Q\\leftrightarrow TQ^{-1}$ symmetry of the\ntriply graded Khovanov-Rozansky homology, and in addition satisfies a number of\npredictions coming from a conjectural connection with Hilbert schemes of points\nin the plane. We compute this invariant for all positive powers of the full\ntwist and match it to the family of ideals appearing in Haiman's description of\nthe isospectral Hilbert scheme.