Matthew Hogancamp

Categorified Young symmetrizers and stable homology of torus links II

We construct complexes

An exceptional collection for Khovanov homology

P_{1^{n}}

On the computation of torus link homology

P1n of Soergel bimodules which categorify the Young idempotents corresponding to one-column partitions. A beautiful recent conjecture (Flag Hilbert schemes, colored projectors and Khovanov–Rozansky homology. arXiv:1608.07308, 2016) of Gorsky–Neguț–Rasmussen relates the Hochschild homology of categorified Young idempotents with the flag Hilbert scheme. We prove this conjecture for

A polynomial action on colored sl(2) link homology

P_{1^{n}}

Idempotents in triangulated monoidal categories

P1n and its twisted variants. We also show that this homology is also a certain limit of Khovanov–Rozansky homologies of torus links. Along the way we obtain several combinatorial results which could be of independent interest.

An involutive upsilon knot invariant

The SO(3) Kauffman polynomial and the chromatic polynomial of planar graphs are categorified by a unique extension of the Khovanov homology framework. Many structural observations and computations of homologies of knots and spin networks are included.