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 spirit of Bar Natan's construction of Khovanov homology, we give a categorification of the Vandermonde determinant. Given a sequence of positive integers, we construct a commutative diagram in the shape of the Bruhat order whose nodes are colored smoothings of a 2-strand torus link and whose arrows are colored cobordisms. An application of a TQFT to this diagram yields a chain complex whose Euler characteristic is the Vandermonde determinant evaluated at the sequence of positive integers we started with. A generalization to arbitrary link diagrams is given, producing categorifications of certain generalized Vandermonde determinants. We also address functoriality of this construction.