Related papers: Closed oriented 3-manifolds are subtle equivalence…
We prove a classification theorem for transitive Anosov and pseudo-Anosov flows on closed 3-manifolds, up to orbit equivalence. In many cases, flows on a 3-manifold $M$ are completely determined by the set of free homotopy classes of their…
By classical results of Rochlin, Thom, Wallace and Lickorish, it is well-known that any two 3-manifolds (with diffeomorphic boundaries) are related one to the other by surgery operations. Yet, by restricting the type of the surgeries, one…
We investigate the disparity between smooth and topological almost concordance of knots in general 3-manifolds Y. Almost concordance is defined by considering knots in Y modulo concordance in Yx[0,1] and the action of the concordance group…
One can define the complexity of a smooth 4-manifold as the minimal sum of the number of disks, strands and crossings in a Kirby diagram. Martelli proved that the number of homeomorphism classes of complexity less than n grows as $n^2$. In…
We calculate the homology groups of certain 2-connected 7-manifolds admitting quasi-regular Sasaki-Einstein metrics. These manifolds are links that arise as Thom-Sebastiani sums of chain type singularities and cycle type singularities.…
A classical theorem due to Wadsley implies that, on a connected contact manifold all of whose Reeb orbits are closed, there is a common period for the Reeb orbits. In this paper we show that, for any Reeb flow on a closed connected…
Given a finite point set P in general position in the plane, a full triangulation is a maximal straight-line embedded plane graph on P. A partial triangulation is a full triangulation of some subset P' of P containing all extreme points in…
We describe a construction procedure of infinite sets of $2$-links in closed simply connected 4-manifolds that are topologically isotopic, smoothly inequivalent and componentwise topologically unknotted. These 2-links are the first examples…
Covering moves relate colored link diagrams appearing as the branch sets of simple branched coverings of $S^3$ by the same 3-manifold. We provide a complete set of covering moves on plat closures of braids in each fixed degree $d \geq 4$,…
Given a 3-manifold $Y$ and a free homotopy class in $[S^1,Y]$, we investigate the set of topological concordance classes of knots in $Y \times [0,1]$ representing the given homotopy class. The concordance group of knots in the 3-sphere acts…
The closure of a braid in a closed orientable surface $\Sigma$ is a link in $\Sigma\times S^1$. We classify such closed surface braids up to isotopy and homeomorphism (with a small indeterminacy for isotopy of closed sphere braids),…
Flip graphs are a ubiquitous class of graphs, which encode relations induced on a set of combinatorial objects by elementary, local changes. Skeletons of associahedra, for instance, are the graphs induced by quadrilateral flips in…
The Nielsen Conjecture for Homeomorphisms asserts that any homeomorphism $f$ of a closed manifold is isotopic to a map realizing the Nielsen number of $f$, which is a lower bound for the number of fixed points among all maps homotopic to…
We prove the finiteness of the number of blow-analytic equivalence classes of embedded plane curve germs for any fixed number of branches and for any fixed value of $\mu'$ ---a combinatorial invariant coming from the dual graphs of good…
A graph is 1-planar if it can be drawn in the plane so that each edge is crossed by at most one another edge. In this work we prove that each 1-planar graph of minimum degree at least $3$ contains an edge with degrees of its endvertices of…
The Milnor degree of a 3-manifold is an invariant that records the maximum simplicity, in terms of higher order linking, of any link in the 3-sphere that can be surgered to give the manifold. This invariant is investigated in the context of…
We present a short proof of the following Pontryagin theorem, whose original proof was complicated and has never been published in details: {\bf Theorem.} Let $M$ be a connected oriented closed smooth 3-manifold. Let $L_1(M)$ be the set of…
We define a new combinatorial class of triangulations of closed 3-manifolds, satisfying a weak version of 0-efficiency combined with a weak version of minimality, and study them using twisted squares. As an application, we obtain strong…
$2$-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.
A knot (or link) diagram is said to be everywhere equivalent if all the diagrams obtained by switching one crossing represent the same knot (or link). We classify such diagrams of a closed 3-braid.