arXiv research

New spines found in complex 4D shapes.

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

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

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

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

A celebrated result concerning triangulations of a given closed 3-manifold is that any two triangulations with the same number of vertices are connected by a sequence of so-called 2-3 and 3-2 moves. A similar result is known for ideal triangulations of topologically finite non-compact 3-manifolds. These results build o…

Paper proves unique contact structure supported by positive flow-spines.

Nonpositive towers property in 3-manifolds spines.

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.

Bing's house-like spines approximate all PL manifolds.

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.

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…

Knotted trivalent graphs (KTGs) form a rich algebra with a few simple operations: connected sum, unzip, and bubbling. With these operations, KTGs are generated by the unknotted tetrahedron and Moebius strips. Many previously known representations of knots, including knot diagrams and non-associative tangles, can be tur…

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

New geometric spine for Artin groups defined by cube complexes.

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.

Study filling links in 3-manifolds to understand their topological properties.

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.

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

In math.GT/0106017 it was shown that thin position on Heegaard spines can be a useful tool for analyzing the topology of knots in 3-space. The proof there (specifically, of the Goda-Teragaito conjecture) requires masses of technical detail; it is easy to lose track of the underlying ideas. The present paper gives an ov…

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…

$2$-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.

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

New proof confirms surfaces can be divided into polygons.

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 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…

New applications of trace embedding lemma show exotic 4-manifolds properties.

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…

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 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 …

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…

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…

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

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…

New analysis of crushing surfaces of positive genus impacts triangulation complexity.