Graph manifolds, left-orderability and amalgamation

Free Products with Amalgamation and 3-manifolds

Heegaard Splittings of Sufficiently Complicated 3-manifolds Ii: Amalgamation

Let M1 and M2 be compact, orientable 3-manifolds, and M the manifold obtained by gluing some component F of ∂M1 to some component of ∂M2 by a homeomorphism φ. We show that when φ is “sufficiently complicated” then (1) the amalgamation of low genus, unstabilized, boundary-unstabilized Heegaard splittings of Mi is an unstabilized splitting of M , (2) every low genus, unstabilized Heegaard splitti...

Three Manifolds and Graph Invariants

We show how the Turaev–Viro invariant can be understood within the framework of Chern–Simons theory with gauge group SU(2). We also describe a new invariant for certain class of graphs by interpreting the triangulation of a manifold as a graph consisiting of crossings and vertices with three lines. We further show, for S3 and RP 3, that the Turaev-Viro invariant is the square of the absolute va...

Graph Coloring Manifolds

We introduce a new and rich class of graph coloring manifolds via the Hom complex construction of Lovász. The class comprises examples of Stiefel manifolds, series of spheres and products of spheres, cubical surfaces, as well as examples of Seifert manifolds. Asymptotically, graph coloring manifolds provide examples of highly connected, highly symmetric manifolds.

Adjacency, Inseparability, and Base Orderability in Matroids

Two elements in an oriented matroid are inseparable if they have either the same sign in every signed circuit containing them both or opposite signs in every signed circuit containing them both. Two elements of a matroid are adjacent if there is no M(K4)-minor using them both, and in which they correspond to a matching of K4. We prove that two elements e, { of an oriented matroid are inseparabl...