Constructingn-Manifolds from Spines

Knot manifolds with isomorphic spines

We study the relation between the concept of spine and the representation of orientable bordered 3-manifolds by Heegaard diagrams. As a consequence, we show that composing invertible non-amphicheiral knots yields examples of topologically different knot manifolds with isomorphic spines. These results are related to some questions listed in [9], [11] and recover the main theorem of [10] as a cor...

Convexity of Morse Stratifications and Gradient Spines of 3-manifolds

We notice that a generic nonsingular gradient field v = ∇f on a compact 3-fold X with boundary canonically generates a simple spine K(f, v) of X. We study the transformations of K(f, v) that are induced by deformations of the data (f, v). We link the Matveev complexity c(X) of X with counting the double-tangent trajectories of the v-flow, i.e. the trajectories that are tangent to the boundary ∂...

Convexity of Morse Stratifications and Spines of 3-manifolds

The gradient fields v of nonsingular functions f on compact 3folds X with boundary are used to generate their spines K(f, v). We study the transformations of K(f, v) that are induced by deformations of the data (f, v). We link the Matveev complexity of c(X) with counting the trajectories of the v-flow that are tangent to the boundary ∂X at a pair of distinct points (call them double-tangent tra...

Spines and Homology of Thin Riemannian Manifolds with Boundary

Branched Spines and Contact Structures on 3-manifolds

The study of the characteristic foliation induced on an embedded surface is one of the main themes in 3-dimensional contact topology. In particular (see [10]) one can describe exactly what foliations arise (“existence results”). Moreover one knows that the characteristic foliation F = Fξ(Σ) determines uniquely the contact structure ξ in a neighbourhood of a surface Σ (“uniqueness results”). (By...