Complexes of not $i$-connected graphs

We define the $k$-cut complex of a graph $G$ with vertex set $V(G)$ to be the simplicial complex whose facets are the complements of sets of size $k$ in $V(G)$ inducing disconnected subgraphs of $G$. This generalizes the Alexander dual of a graph complex studied by Fr\"oberg (1990), and Eagon and Reiner (1998). We describe the effect of various graph operations on the cut complex, and study its shellability, homotopy type and homology for various families of graphs, including...

We propose a new method of computing cohomology groups of spaces of knots in $\R^n$, $n \ge 3$, based on the topology of configuration spaces and two-connected graphs, and calculate all such classes of order $\le 3.$ As a byproduct we define the higher indices, which invariants of knots in $\R^3$ define at arbitrary singular knots. More generally, for any finite-order cohomology class of the space of knots we define its principal symbol, which lies in a cohomology group of a ...

For positive integers k,n, we investigate the simplicial complex NM_k(n) of all graphs G on vertex set [n] such that every matching in G has size less than k. This complex (along with other associated cell complexes) is found to be homotopy equivalent to a wedge of spheres. The number and dimension of the spheres in the wedge are determined, and (partially conjectural) links to other combinatorially defined complexes are described. In addition we study for positive integers r...

The purpose of this paper is to provide methods for determining the associated primes of the square of the Alexander dual of the edge ideal for an m-hypergraph H. We prove a general method for detecting associated primes of the square of the Alexander dual of the edge ideal based on combinatorial conditions on the m-hypergraph. Also, we demonstrate a more efficient combinatorial criterion for detecting the non-existence of non-minimal associated primes. In investigating 3-hyp...

In this article, we consider the bipartite graphs $K_2 \times K_n$. We first show that the connectedness of $\mathcal{N}(K_{n+1}^{K_n}) =0$. Further, we show that $\text{Hom}(K_2 \times K_{n}, K_{m})$ is homotopic to $S^{m-2}$, if $2\leq m <n$.

Hypergraphs have seen widespread applications in network and data science communities in recent years. We present a survey of recent work to define topological objects from hypergraphs -- specifically simplicial, relative, and chain complexes -- that can be used to build homology theories for hypergraphs. We define and describe nine different homology theories and their relevant topological objects. We discuss some interesting properties of each method to show how the hypergr...

In the paper we prove that the primitive part of the Sinha homology spectral sequence E^2-term for the space of long knots is rationally isomorphic to the homotopy E^2-term. We also define natural graph-complexes computing the rational homotopy of configuration and of knot spaces.

In this paper we enumerate and classify the ``simplest'' pairs (M,G) where M is a closed orientable 3-manifold and G is a trivalent graph embedded in M. To enumerate the pairs we use a variation of Matveev's definition of complexity for 3-manifolds, and we consider only (0,1,2)-irreducible pairs, namely pairs (M,G) such that any 2-sphere in M intersecting G transversely in at most 2 points bounds a ball in M either disjoint from G or intersecting G in an unknotted arc. To c...

Hypergraph is a topological model for networks. In order to study the topology of hypergraphs, the homology of the associated simplicial complexes and the embedded homology have been invented. In this paper, we give some algorithms to compute the homology of the associated simplicial complexes and the embedded homology of hypergraphs as well as some heuristics for efficient computations.

We introduce a notion of intrinsic linking and knotting for virtual spatial graphs. Our theory gives two filtrations of the set of all graphs, allowing us to measure, in a sense, how intrinsically linked or knotted a graph is; we show that these filtrations are descending and non-terminating. We also provide several examples of intrinsically virtually linked and knotted graphs. As a byproduct, we introduce the {\it virtual unknotting number} of a knot, and show that any knot ...