arXiv research

We establish a calculus for branched spines of 3-manifolds by means of branched Matveev-Piergallini moves and branched bubble-moves. We briefly indicate some of its possible applications in the study and definition of State-Sum Quantum Invariants.

We introduce and analyze the characteristic foliation induced by a contact structure on a branched surface, in particular a branched standard spine of a 3-manifold. We extend to (fairly general) singular foliations of branched surfaces the local existence and uniqueness results which hold for genuine surfaces. Moreover…

Simplified combinatorial descriptions of branched spines for 3-manifolds using primary MP move and sliding moves.

$2$-stratifolds are a generalization of $2$-manifolds in that there are disjoint simple closed branch curves. We obtain a list of all closed $3$-manifolds that have a $2$-stratifold as a spine.

Paper proves S-stability of foliations on flow-spines with transverse Reeb flow.

We define and study branched shadows of 4-manifolds as a combination of branched spines of 3-manifolds and Turaev's shadows. We use these objects to combinatorially represent 4-manifolds equipped with $Spin^c$-structures and homotopy classes of almost complex structures. We then use branched shadows to study complex 4-…

We give upper bounds of the Matveev complexities of two-bridge link complements by constructing their spines explicitly. In particular, we determine the complexities for an infinite sequence of two-bridge links corresponding to the continued fractions of the form [2,1,...,1,2]. We also give upper bounds for the 3-manif…

We provide combinatorial realizations, according to the usual objects/moves scheme, of the following three topological categories: (1) pairs (M,v) where M is a 3-manifold (up to diffeomorphism) and v is a (non-singular vector) field, up to homotopy; here possibly the boundary of M is non-empty and v may be tangent to t…

The paper constructs symplectic forms on 4-manifolds using branched coverings and holomorphic line bundles.

Nonpositive towers property in 3-manifolds spines.

A flow-spine of a 3-manifold is a spine admitting a flow that is transverse to the spine, where the flow in the complement of the spine is diffeomorphic to a constant flow in an open ball. We say that a contact structure on a closed, connected, oriented 3-manifold is supported by a flow-spine if it has a contact form w…

In this paper we raise the question whether every closed Riemannian manifold has a spine of minimal area, and we answer it affirmatively in the surface case. On constant curvature surfaces we introduce the spine systole, a continuous real function on moduli space that measures the minimal length of a spine in each surf…

Unique simple spines of homotopy 2-spheres are shown to be ambiently isotopic.

We show that infinitely many of the simply connected 4-manifolds constructed by Levine and Lidman that do not admit PL spines actually admit topological spines.

Corrects a 1-off error in Harer's spine dimension calculation for decorated Teichmüller spaces.

Constructs Teichmüller curve to study Thurston spine structure.

This paper proves a map from flow-spines to contact structures is surjective.

The Thurston spine's properties are studied in relation to Morse-Smale complexes.

In this article we establish the relation between the spines of 3-manifolds and the polyhedra with identified faces. We do this by showing that the spines of the closed, connected, orientable 3-manifolds can be presented through polyhedra with identified faces in a very natural way. We also prove the equivalence betwee…

A special spine of a three-manifold is said to be poor if it does not contain proper simple subpolyhedra. Using the Turaev-Viro invariants, we establish that every compact three-dimensional manifold M with connected nonempty boundary has a finite number of poor special spines. Moreover, all poor special spines of the m…

The abstract extends Reidemeister theorem to 3-manifolds using diagrams of links and bands.

New geometric spine for Artin groups defined by cube complexes.

The paper studies geometric structures of polynomial spaces.

Paper shows mapping class group-equivariant Teichmüller space deformation to Thurston spine.

Thurston's spine dimension exceeds virtual cohomological dimension.

A spine is constructed for a non-orientable surface's decorated Teichmüller space.

New 3-manifold spines with unique Whitehead graphs identified.

Equivariant trisections for group actions on 4-manifolds are introduced and studied.

The paper shows conditions under which certain 4-manifolds have no smooth spines.

Theory of link projections to 3-manifold spines, proving combinatorial moves for isotopic links.

Any bounding compact smooth manifold bounds a compact manifold with a spine consisting of transversely intersecting codimension one submanifolds. This paper provides details for a picture proof given in previous papers with S. Akbulut.

We show that all PL manifolds of dimension $\geq 3$ have spines similar to Bing's house with two rooms. Beyond this we explore approximation rigidity and an $h$-principle.

Outer space and Teichmüller space fail well-rounded retract analogy.

Geometrical spines are defined for 3-manifolds with natural metrics, in particular, for lens manifolds. We show that any spine of L(p,q) close enough to its geometrical spine (i.e., to the cut locus with respect to the standard metric) contains at least E(p,q)-3 vertices, which is exactly the conjectured value for Matv…

New proof confirms surfaces can be divided into polygons.

The Brieskorn manifolds $B(p,q,r)$ are the $r$-fold cyclic coverings of the 3-sphere $S^{3}$ branched over the torus knot $T(p,q)$. The generalised Sieradski groups $S(m,p,q)$ are groups with $m$-cyclic pre\-sen\-tation $G_{m}(w)$, where defining word $w$ has a special form, depending of $p$ and $q$. In particular, $S(…

Continuum-wise hyperbolicity is exactly the pseudo-Anosov dynamics with spine singularities.

The paper studies curves in surfaces using flow-spines and apparent contours.

New contractible complex shows virtual cohomological dimension of RAAGs.

For a 3-dimensional manifold $M^3$, its complexity $c(M^3)$, introduced by S.Matveev, is the minimal number of vertices of an almost simple spine of $M^3$; in many cases it is equal to the minimal number of tetrahedra in a singular triangulation of $M^3$. An approach to estimating $c(M^3)$ from below for total spaces o…

We show that under reasonable conditions, the spines of the handlebodies of a strongly irreducible Heegaard splitting will intersect a closed ball in a graph which is isotopic into the boundary of the ball. This is in some sense a generalization of the results by Scharlemann on how a strongly irreducible Heegaard split…

We construct infinitely many smooth 4-manifolds which are homotopy equivalent to $S^2$ but do not admit a spine, i.e., a piecewise-linear embedding of $S^2$ which realizes the homotopy equivalence. This is the remaining case in the existence problem for codimension-2 spines in simply-connected manifolds. The obstructio…

We introduce a novel approach for predicting the progression of adolescent idiopathic scoliosis from 3D spine models reconstructed from biplanar X-ray images. Recent progress in machine learning have allowed to improve classification and prognosis rates, but lack a probabilistic framework to measure uncertainty in the …

Study on symmetric automorphisms of RAAGs, proving finiteness properties and contractibility.

We prove that for any contact 3-manifold supported by a spinal open book decomposition with planar pages, there is a universal bound on the Euler characteristic and signature of its minimal symplectic fillings. The proof is an application of the spine removal surgery operation recently introduced in joint work of the a…

A canal surface is the envelope of a moving sphere with varying radius, defined by the trajectory C(t) (spine curve) of its center and a radius function r(t). In this paper, we investigate when parameter curves of the canal surface are also lines of curvature. Last of all, for special spine curves we obtain the radius …

Let T_n be the Teichmueller space of flat metrics on the n-dimensional torus and identify SL(n,Z) with the corresponding mapping class group. We prove that the subset Y consisting of those points at which the systoles generate the fundamental group of the torus is, for n > 4, not contractible. In particular, Y is not a…

Study of Penner's cocycle on fatgraph complex.