Related papers: Roots in 3-manifold topology

We prove that for any knot $K$, there exists a one-vertex triangulation of the $3$-sphere containing an edge forming $K$. The proof is constructive, and based on fully augmented links. We use our method to produce ``complicated'' simplicial…

We prove that, if $m$ is sufficiently large, every graph on $m+1$ vertices that has a universal vertex and minimum degree at least $\lfloor \frac{2m}{3} \rfloor$ contains each tree $T$ with $m$ edges as a subgraph. Our result confirms, for…

It is shown that for any locally knotted edge of a 3-connected graph in $S^3$, there is a ball that contains all of the local knots of that edge and is unique up to an isotopy setwise fixing the graph. This result is applied to the study of…

We give closed form expressions for the numbers of multi-rooted plane trees with specified degrees of root vertices. This results in an infinite number of integer sequences some of which are known to have an alternative interpretation. We…

This is the second of a series of papers studying real algebraic threefolds using the minimal model program. The main result is the following. Let $X$ be a smooth projective real algebraic 3-fold. Assume that the set of real points is an…

We establish sufficient conditions for the C$^*$-simplicity of two classes of groups. The first class is that of groups acting on trees, such as amalgamated free products, HNN-extensions, and their normal subgroups; for example normal…

In 3-dimensional manifolds, we prove that generically in$Diff^1_m(M)$, the existence of a minimal expanding invariant foliation implies stable Bernoulliness.

We prove that finding a rooted subtree with at least $k$ leaves in a digraph is a fixed parameter tractable problem. A similar result holds for finding rooted spanning trees with many leaves in digraphs from a wide family $\cal L$ that…

We will describe some results regarding the algorithmic nature of homeomorphism problems for manifolds; in particular, the following theorem. Theorem 1: Every PL or smooth simply connected manifold M^n of dimension n at least 5 can be…

Using a notation of corner between edges when graph has a fixed rotation, i.e. cyclical order of edges around vertices, we define combinatorial objects - combinatorial maps as pairs of permutations, one for vertices and one for faces.…

In this paper, we study functions of the roots of a univariate polynomial in which the roots have a given multiplicity structure $\mu$. Traditionally, root functions are studied via the theory of symmetric polynomials; we extend this theory…

The division of compact Riemann surfaces into 3 cases K_C<0, g=0, or K_C=0, g=1, or K_C>0, g>=2 is well known, and corresponds to the familiar trichotomy of spherical, Euclidean and hyperbolic non-Euclidean plane geometry. Classification…

We study the first homology group of the Milnor fiber of sharp arrangements in the real projective plane. Our work relies on the minimal Salvetti complex of the deconing arrangement and its boundary map. We describe an algorithm which…

This paper contains three main results. Firstly, we give an elementary proof of the following statement: Let $M$ be a (closed, in both the geometrical and topological sense of the word) topological manifold embedded in $\mathbb{R}^d$. If…

The paper studies the question of existence of polynomials with given roots over associative non-commutative rings with identity. It is shown that in the case of an associative division ring for arbitrary n elements of this ring there…

We propose a new arithmetic for non-empty rooted unordered trees simply called trees. After discussing tree representation and enumeration, we define the operations of tree addition, multiplication and stretch, prove their properties, and…

We study continuity of the roots of nonmonic polynomials as a function of their coefficients using only the most elementary results from an introductory course in real analysis and the theory of single variable polynomials. Our approach…

This is a draft of a book submitted for publication by the AMS. Its theme is the remarkable interplay, accelerating in the last few decades, between topology and the theory of orderable groups, with applications in both directions. It…

We define a concept which we call multiplicity. First, multiplicity of a morphism is defined. Then the multiplicity of an object over another object is defined to be the minimum of the multiplicities of all morphisms from one to another.…

To each isolated critical point of a smooth function on a 3-manifold we put in correspondence a tree (graph without cycles). We will prove that functions are topologically equivalent in the neighborhoods of critical points if and only if…