Using the tools of algebraic Morse theory and the thin poset approach to constructing homology theories, we give a categorification of Whitney’s broken circuit theorem for the chromatic polynomial and Stanley’s chromatic symmetric function. As an application, we obtain bounds on the homological span for both homology theories.

In this paper, we investigate the strength of chromatic symmetric homology as a graph invariant. Chromatic symmetric homology is a lift of the chromatic symmetric function for graphs to a homological setting, and its Frobenius characteristic is a q,t generalization of the chromatic symmetric function. We exhibit three pairs of graphs where each pair has the same chromatic symmetric function but distinct homology. We also show that integral chromatic symmetric homology contains torsion and, based on computations, conjecture that torsion in bigrading (1,0) detects nonplanarity in the graph.

Motivated by generalizing Khovanov’s categorification of the Jones polynomial, we study functors from thin posets to abelian categories. Such functors produce cohomology theories, and we find that CW posets–face posets of regular CW complexes–satisfy conditions making them particularly suitable for the construction of such cohomology theories. We consider a category whose objects are tuples (P,A,F,c) where P is a CW poset, A is an abelian category, F is a functor from P to A, and c is a certain coloring of the Hasse diagram of P making intervals of length 2 anticommute. We show the cohomology arising from a tuple (P,A,F,c) is functorial, and independent of the coloring c up to natural isomorphism. Such a construction provides a framework for the categorification of a variety of familiar topological/combinatorial invariants, anything expressible as a rank alternating sum over a thin poset.

In the integral Khovanov homology of links, the presence of odd torsion is rare. Homologically thin links–links whose Khovanov homology is supported on two adjacent diagonals–are known to only contain torsion of order 2. In this paper, we prove a local version of this result. If the Khovanov homology of a link is supported in two adjacent diagonals over a range of homological gradings and the Khovanov homology satisfies some other mild restrictions, then the Khovanov homology of that link has only torsion of order 2 over that range of homological gradings. These conditions are then shown to be met by an infinite family of 3-braids, strictly containing all 3-strand torus links, thus giving a partial answer to Sazdanovic and Przytycki’s conjecture that 3-braids have only torsion of order 2 in Khovanov homology. We also give explicit computations of integral Khovanov homology for all links in this family.

This is an expository chapter in Springer’s “Introduction to Soergel Bimodules”. In classical linear algebra, given a diagonalizable operator on a vector space, Lagrange interpolation produces an idempotent decomposition of the identity corresponding to the projections to eigenspaces. In this chapter, we explain a categorical analogue of this procedure due to Elias and Hogancamp. Given a “diagonalizable” functor acting on a monoidal homotopy category, we produce idempotent functors which project to “eigencategories”. The main application is to the full twist Rouquier complex acting on the homotopy category of Soergel bimodules.