Anton Dochtermann

Tropical types and associated cellular resolutions

Abstract An arrangement of finitely many tropical hyperplanes in the tropical torus T d − 1 leads to a notion of ‘type’ data for points in T d − 1 , with the underlying unlabeled arrangement giving rise to ‘coarse type’. It is shown that the decomposition of T d − 1 induced by types gives rise to minimal cocellular resolutions of certain associated monomial ideals. Via the Cayley trick from geometric combinatorics this also yields cellular resolutions supported on mixed subdivisions of dilated simplices, extending previously known constructions. Moreover, the methods developed lead to an algebraic algorithm for computing the facial structure of arbitrary tropical complexes from point data.

The universality of Hom complexes of graphs

It is shown that given a connected graph T with at least one edge and an arbitrary finite simplicial complex X, there is a graph G such that the complex Hom(T,G) is homotopy equivalent to X. The proof is constructive, and uses a nerve lemma. Along the way several results regarding Hom complexes, exponentials of graphs, and subdivisions are established that may be of independent interest.

GRAPHS WITH DISJOINT LINKS IN EVERY SPATIAL EMBEDDING

We exhibit a graph, G12, that in every spatial embedding has a pair of non-splittable 2 component links sharing no vertices or edges. Surprisingly, G12 does not contain two disjoint copies of graphs known to have non-splittable links in every embedding. We exhibit other graphs with this property that cannot be obtained from G12 by a finite sequence of Δ-Y and/or Y-Δ exchanges. We prove that G12 is minor minimal in the sense that every minor of it has a spatial embedding that does not contain a pair of non-splittable 2 component links sharing no vertices or edges.

Hom complexes and homotopy in the category of graphs

Abstract In this extended abstract we develop a notion of ×-homotopy of graph maps that is based on the internal hom associated to the categorical product. We show that graph ×-homotopy is characterized by the topological properties of the so-called Hom complex, a functorial way to assign a poset to a pair of graphs. Along the way we establish some structural properties of Hom complexes involving products and exponentials of graphs, as well as a symmetry result which can be used to reprove a theorem of Kozlov involving foldings of graphs. We end with a discussion of graph homotopies arising from other internal homs, including the construction of ‘ A -theory’ associated to the cartesian product in the category of reflexive graphs. For proofs and further discussions we refer the reader to the full paper [Anton Dochtermann. Hom complexes and homotopy theory in the category of graphs. arXiv:math.CO/0605275 ].