Related papers: Homotopy on spatial graphs and the Sato-Levine inv…
Topologists are sometimes interested in space-valued diagrams over a given index category, but it is tricky to say what such a diagram even is if we look for a notion that is stable under equivalence. The same happens in (homotopy) type…
We prove the existence of a polynomial invariant that satisfies the HOMFLY skein relation for links in a lens space. In the process we also develop a skein theory of toroidal grid diagrams in a lens space.
We develop a theory of $\times$-homotopy, fundamental groupoids and covering spaces that apply to non-simple graphs, generalizing existing results for simple graphs. We prove that $\times$-homotopies from finite graphs can be decomposed…
The Surjective Homomorphism problem is to test whether a given graph G called the guest graph allows a vertex-surjective homomorphism to some other given graph H called the host graph. The bijective and injective homomorphism problems can…
An explicit polynomial in the linking numbers $l_{ij}$ and Milnor's triple linking numbers $\mu(rst)$ on six component links is shown to be a well-defined finite type link-homotopy invariant. This solves a problem raised by B. Mellor and D.…
We propose to study homomorphisms of connectome graphs. Homomorphisms can be studied as sequences of elementary homomorphisms - folds, which identify pairs of vertices. Several fold types are defined. Initial computation results for some…
An invariant for cospectral graphs is a property shared by all cospectral graphs. In this paper, we establish three novel arithmetic invariants for cospectral graphs, revealing deep connections between spectral properties and combinatorial…
We introduce a spatial graph and hypergraph model that smoothly interpolates between a graph with purely pairwise edges and a graph where all connections are in large hyperedges. The key component is a spatial clustering resolution…
We construct a cobordism group for embedded graphs in two different ways, first by using sequences of two basic operations, called "fusion" and "fission", which in terms of cobordisms correspond to the basic cobordisms obtained by attaching…
Using a ribbon structure of the graph, we construct a dissection of the symmetric edge polytope of a graph into unimodular simplices. Our dissection is shellable, and one can interpret the elements of the resulting $h$-vector via graph…
New homotopy approach to the analysis of nonlinear higher-spin equations is developed. It is shown to directly reproduce the previously obtained local vertices. Simplest cubic (quartic in Lagrangian nomenclature) higher-spin interaction…
Graph classification plays an important role is data mining, and various methods have been developed recently for classifying graphs. In this paper, we propose a novel method for graph classification that is based on homotopy equivalence of…
In this paper we consider aspects of geometric observability for hypergraphs, extending our earlier work from the uniform to the nonuniform case. Hypergraphs, a generalization of graphs, allow hyperedges to connect multiple nodes and…
A geometric graph is a simple graph G together with a straight line drawing of G in the plane with the vertices in general position. Two geometric realizations of a simple graph are geo-isomorphic if there is a vertex bijection between them…
This article presents a survey of some recent results in the theory of spatial graphs. In particular, we highlight results related to intrinsic knotting and linking and results about symmetries of spatial graphs. In both cases we consider…
We initiate the study of classical knots through the homotopy class of the n-th evaluation map of the knot, which is the induced map on the compactified n-point configuration space. Sending a knot to its n-th evaluation map realizes the…
We define several homology theories for central hyperplane arrangements, categorifying well-known polynomial invariants including the characteristic polynomial, Poincare polynomial, and Tutte polynomial. We consider basic algebraic…
This is a survey article on the stable cohomotopy refinement of Seiberg-Witten invariants containing also new results, for example: - Stable cohomotopy groups describe path components of certain mapping spaces. - Relation of stable…
We introduce a crossed module of piecewise linear surfaces and study the signature homomorphism, defined as the surface holonomy of a universal translation invariant $2$-connection. This provides a transform whereby surfaces are represented…
Knots and links in 3-manifolds are studied by applying intersection invariants to singular concordances. The resulting link invariants generalize the Arf invariant, the mod 2 Sato-Levine invariants, and Milnor's triple linking numbers.…