Combinatorial and topological aspects of path posets, and multipath cohomology

Multipath cohomology is a cohomology theory for directed graphs, which is defined using the path poset. The aim of this paper is to investigate combinatorial properties of path posets and to provide computational tools for multipath cohomology. In particular, we develop acyclicity criteria and provide computations of multipath cohomology groups of oriented linear graphs. We further interpret the path poset as the face poset of a simplicial complex, and we investigate realisability problems

On the homotopy type of multipath complexes

A multipath in a directed graph is a disjoint union of paths. The multipath complex of a directed graph G is the simplicial complex whose faces are the multipaths of G. We compute Euler characteristics, and associated generating functions, of the multipath complexes of directed graphs from certain families, including transitive tournaments and complete bipartite graphs. We show that if G is a linear graph, polygon, small grid or transitive tournament, then the homotopy type of the multipath complex of G is always contractible or a wedge of spheres. We introduce a new technique for decomposing directed graphs into dynamical regions, which allows us to simplify the homotopy computations

A note on the weak splitting number

The weak splitting number $wsp(L)$ of a link $L$ is the minimal number of crossing changes needed to turn $L$ into a split union of knots. We describe conditions under which certain $\mathbb{R}$-valued link invariants give lower bounds on $wsp(L)$. This result is used both to obtain new bounds on $wsp(L)$ in terms of the multivariable signature and to recover known lower bounds in terms of the $\tau$ and $s$-invariants. We also establish new obstructions using link Floer homology and apply all these methods to compute $wsp$ for all but two of the $130$ prime links with $9$ or fewer crossings